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Dragons are often depicted with a set of plate-like scales along their belly like a snake. In snakes, this adaptation is to aid in their slithering locomotion. However, dragons don't slither; they have large legs and wings for walking and flying
Is there some other function that could justify snake-like scales on the belly of a quadruped like a dragon?
The same reason for which the crocodiles have plated bellies, in spite of having legs able to walk - the conformation of their legs aren't keeping the body clean of the ground all the time, on the contrary - most of the time their "walk" resemble crawling more than actual walking.
Even more so as the "usual" habitat in which the dragons are depicted are mountains with sharp peaks - denoting a material that is hard, brittle and prone to fragment in shards (medium grained oriented in such a way that the rock is easily split into thin flakes or plates in geological parlance. Or some other forms of volcanic origin).
If you like this better: evolution favoured dragons with plated bellies simply because the individuals which didn't get one were slain by heroes before they were able to reproduce.
The simple answer is protection. For one, those heroic dragonslayers running around would dearly love a dragon with a vulnerable belly: one good stab with a spear, and the dragon is probably dead. This is doubly relevant when one considers that with the typical large dragons, the stomach is the easiest target to reach and (depending on size) possibly the only target that can reasonably be struck with melee weapons, unless the dragon is so generous as to wait while the slayers grab a ladder, prop it up against the dragon like they're about to do some work on the roof, climb up towards the head or back... no, I really don't see any sane dragon being that nice. Having scales on the belly would actually be more important than most other points on the body.
If you want to stick with evolution-based reasons from before humans come into the picture, the logic doesn't really change. Belly scales shield one from any rude animals stupid enough to try and ambush the dragon (like some killer venomous snake, ironically enough), or as Adrian Colomitchi notes in his answer, dangerous terrain. It would admittedly be hilarious if a dragon slipped when a rock cracked under its foot and impaled itself on a stalagmite or something, but in general any dragons trying to use potentially cramped caves or tunnels are going to want scales to avoid gashing themselves frequently on any sharp rocks; it doesn't have to be fatal to be a serious concern.
If your dragons are prone to fighting with one another (or if there are other creatures big and nasty enough to compete with them, like griffons or giant eagles or some such), belly scales also shield them from the claws of those enemies. Gut wounds are often at least temporarily crippling if they get past the outer tissues, and in nature you don't typically have the luxury of antibiotics, time to recover fully from disabling injuries, etc.; a predator with a wound like that is most likely dead within the week, since a torn liver or stomach is definitely going to spoil its hunting prowess, setting it up to starve to death unless it's very lucky. A layer of scales on the belly to match those on the rest of the body (I'm assuming the standard scaled dragon here) would be only natural to have. Bonus points if dragon courtship rituals tend to the savage, requiring protection just to keep from maiming one or both dragons, but that's probably redundant on top of all the other reasons.
Frankly, dragons seem likely to have scales on their bellies if they have any scales at all. The better question to ask, in fact, is why so many dragons have scales everywhere but on the area in greatest need of them!
Stegosaurs have flexible necks so armor plates have to be small, but if dragons are like dinosaurs their torso is rigid so armor can be as big as possible. You see this in crocodiles which can be attacked from below, armor plates on the belly are as large as possible without impairing movements. An animal that can fly is going to face many attacks from below, it may even be more heavily armored on the belly than the back.
Dragons fly[citation not needed], and any observer can see that the wings of the average dragon are not going to cut the mustard for unassisted flight. Dragons also breathe fire[citation not needed], which requires a source of ignitable liquid or gas. Both these functions are served by the dragon's digestive system, which releases hydrogen from its food and stores it in a sac in its abdomen. The sac provides bouyancy to compensate for the dragon's weight/wingspan deficiency and also fuel for its fire.
This sac will, obviously, inflate extremely rapidly after a substantial meal of, for example, heroes, hobbitses, suitable rocks (trolls) or anything else it can get its claws on and therefore needs to be constrained by something even tougher than regular dragon skin; also the sac must not be unexpectedly vented (by, for example, a lance or arrow) unless the dragon really wants to risk cooking itself and everything else within a hundred yards or so. Both these requirements have led to overlapping armoured plates around the dragon's belly, all the way up to its throat and mouth, which is usually the ultimate discharge point for the hydrogen (flatulence is only used for short-term acceleration; an after-burner, if you like).
Simple evolution in action, nothing to see here.
Dragons normally show a complete lack of respect to the square cube law by existing as they're portrayed, so by all means, a dragon does not conform with the laws of this world, being able to seemingly ignore the natural constraints of biology through yet unknown means (something many would call "magic") and thus don't need to worry about the problems that should normally come from being too heavy. for that reason, being more well protected is inevitably an advantage for them, especially in a place like the belly, where, as far as other living creatures go by, tends to be a fairly vulnerable and important spot in comparison, especially so for a creature that's usually flying around with its belly always on display to any crazy Enough bowman or ballista operator with a black arrow and a "go get em" attitude (and also incredible strength) to aim and shoot at.
At the end of the day, the various evolutionary approaches to increase survival rate we see in the real world come each at their own price. Being able to fly means you're normally not sturdy when compared to a non-flier of similar size, and your size will be naturally limit so long as you're not willing to sacrifice that. Higher defense usually means more weight and less mobility. A larger size means more protection at the cost of more food, while a smaller body grants better mobility and less defense against larger creatures.
However, from the moment the catch that comes with those trade-offs is flung out the window, the most successful organism will the one that, like a dragon with treasure, hoards them all to become the closest thing to a borderline unkillabe monster. The dragon's incredible success can be tracked essentially to its ability to go against the laws that bind every other organism, since they can afford to be big, tough, agile, still able to fly, energy-efficient and disproportionately strong, because all of those cost exactly zero evolutionary drawbacks to it. These things only need to worry about starving to death, diseases and old age, and that middle one assumes they don't also have a superb immune system like some creatures on earth.
Summing up: why would they? Because they can. The dragon's negation factor means it's own evolution of is much closer to the evolution of technology and warfare than it is to normal biological evolution. You're either stacking up more capabilities and further developing those you already have like a powergamer's character or you'll end outcompeted by others of your kind.
In J.R.R. Tolkien's legendarium dragons have soft unarmored bellies and so some of the largest and most fearsom dragons were killed by puny men stabbing them in the stomach area or shooting them in the breast area.
That is how some of the most famous dragon slayings in the legendarium happened.
And I have reason to believe that those dragons were so large that when they stood up straight on their legs their bellies would have seen so high above the ground that someone would have had to jab straight upwards with a long spear to reach them.
And maybe Tolkien's dragons always crawled along the ground in battle, with no space between the ground and their stomachs for any enemy to stab them. So the only places where an enemy could stab or hack would be the sides and the tops of the dragons, protected by armored scales. So maybe Eru could make Morgoth think the dragons didn't need any armor on their unndersides. But of course that is inconsistent with how Azaghal stabbed Glaurung at the Battle of Tears Unnumbered, so apparently Tolkien's dragons didn't aways press their undersides against the ground for protection in battle.
So it always seemed like a major design flaw that Tolkien's dragons had such vulnerable unarmored undersides. How could Morboth have been stupid enough to design dragons with such fatal flaws? And of course the answer could be that Eru made Morgoth fail to see the fatal design flaw, because Eru wanted to make it possible for great heroes to slay dragons.
But if a writer doesn't specifically want to make it possible for a great hero to slay a dragon in his story, because he is telling a diffrent type of story, then they have no need to give their dragons vulnerable, unprotected undersides (or unprotected sides, tops, tiny spots right above vital organs, etc.).
Possibly when a dragon attacks an army, or an army attacks a dragon, in a story, the dragon can walk over the army formations, with their well armoured underside 20, 30, or more feet high, and out of reach of their enemeies, and then suddendly fold their legs and come slamming down on the soldiers below, crushing tens or hundreds to death, with vertical spears and swords snapping like toothpicks. That would crush the hopes that any character might have of being able to stab the dragon in the soft underbelly. And also crush the character if they were under the dragon at the time.
Breeding
War dragons have been selectively bred by warlords and kings, for thousands of years.
War dragons descend from free dragons that had (some) belly armour.
For free dragons, heavy belly armour would only add weight and limit flying. It has hardly any evolutionary advantage. But in a war involving humans and dragons, the dragons have a great advantage, when they have plate-like scales along their belly, it makes them invincible for arrows, spears and swords.
