You need to realize that:
- Humanity is only scratching the surface of understanding evolution. We do not have a scientifically rigorous understanding of how any specific thing could evolve (such as alternative intelligent species).
But we do have a growing understanding of the general pattern.
1: Problem solving opportunities
Evolution is a somewhat random process of small, incremental changes over vast amounts of time. When those changes favor survival, that change survives into the next generation. Therefore, your species need challenges to overcome and problems to solve. Challenges are often passive and require little to no "intelligence." Problems tend to require the evolving species to react to something, and react well. All these can take the form of climate, predator, breeding, eating, and a thousand other opportunities.
2: A window for survival
Predators can't be too successful. The environment can't be too extreme. It's not enough to have problem solving opportunities, those opportunities must also have a "reasonable" chance for success. What do I mean when I quote the word "reasonable?" There isn't a hard-and-fast rule for any particular opportunity that decides whether or not it is (*ahem*) too hot, too cold, or just right. But the window for survival must exist. Is the temperature too hot? Your evolving species must adapt to finding shade — or using water to stay cool. Do the predators have long tongues? Your species must adapt to digging deeper — or learning how to sting the tongue. Did a solar flare raise the temperature too high? Your species is dead. Did the predator develop a longer tongue faster than your species could learn to dig? Your species is dead.
3: A means of surviving or avoiding extinction-level-events
In many ways, Earth's story of evolution is a story of dodging the proverbial bullet. Meteors, exploding calderas, ice ages, shifting plate tectonics, de-oxygenation of the ocean.... life (and, particularly, specific chains of life) had to dodge those bullets to evolve into human intelligence. But, to avoid burning a number on a duplicate, you need a balance just as you do for problem-solving opportunities. You can't avoid all ELEs or you don't weed out the casually successful chains of life from the "superior" chains (and I say "superior" tongue-in-cheek... we created the Austin Powers movies after all...). So it's not the events you're looking to minimize, its the strengths of your evolving life that you're looking to support and propagate.
4: Spread like a
virus successful species
Back in college I had to write a program called "sharks vs. fish." Maybe they still have people write it today to study recursive subroutines. (How far back was that? I wrote it in Fortran....) While the point of the exercise was recursion, the goal was to write a program that simulated the predator-vs-prey relationship. Predators propagate slowly and only in relation to the food supply. Prey propagates like the proverbial rabbits. Predators consume prey on a logarithmic scale (the greater the numbers of predators, the faster they eat the prey). What's the point of all this? Your species must propagate fast enough to ensure there is a next generation. All those evils propounded by anti-human-population-and-expansion activists? Yeah, they're the evolutionary strengths that got us to the point where someone could complain about it. Without those strengths, we wouldn't have made it to the point of complaining about it. However, you can't spread too fast or you no longer have the most powerful compulsion for change: predation. Why have mosquitoes never evolved beyond what they are? Because there's no need to change. They are completely successful in their niche.
5: Availability of dietary requirements
The evolutionary chain that brought the world Humanity in all its glory included the foods necessary for a changing diet brought about by the changes necessary for our evolution. Over-simplifying a LOT, we started with eating oceanic organics, graduated to leaves and grass, moved on to fruits and nuts, then came the need for proteins and fat-rich food sources (yeah... meat..., especially bone marrow), until we ended up today with the ability to complain about people who frequent all-you-can-eat BBQ. Your birds are most likely fruit and nut eaters, maybe a bit of meal worm... but as they evolve, they'll need access to foods that can fuel those changes — notably that can fuel the changes in the brain. But that also means they must change to handle consuming, processing, and expelling those kinds of foods.
Now for the real problem...
At the beginning I said that evolution is a somewhat random process of small, incremental changes over vast amounts of time. It is. The problem is that you're likely looking for a magical summary of the top five events that justify human intelligence and, thus, could justify avian intelligence (or similar). The problem is that it doesn't work that way.
When I say "small" and "vast" I really mean it. Just contemplating how the human eye (and the ability to "see, involving nerves and the brain") evolved is breathtaking. So many eons making small, almost meaningless changes until, boom! Michelangelo.
In other words, no matter what any of us say in response to this question, you're stuck with a worldbuilding problem. If you're trying to map out a "realistic" evolution of anything, you're kinda doomed to failure. Lots of small, incremental changes over vast amounts of time. You'll be stuck with creating a (very) short list of "big events" that will rationalize your end result — an avian species both sentient and sapient. And what, specifically, that list is has more to do with storybuilding (the needs of your story) than it does worldbuilding (the rules underlying the operation of your world). Thus, I've given you the outline of a bunch of potential rules — but it'll be up to you to craft the specifics that your story needs. (I'm assuming a story, I apologize if that's not the case.)
Don't feel bad about that! Worldbuilding is often the art of simplifying things. Even massive worldbulding projects like Orion's Arm are, for all their cool and amazing glory, massive simplifications. Just remember, you should be looking to rationalize your creation, not fully explain it. You'll sleep better if you keep that in mind. Cheers.
One last thing...
If what you're looking for is something that looks like a modern parrot who can competently discuss quantum mechanics with Stephen Hawking, you're going to need to change your paradigm. The changes that must occur to the bird you see sitting on a branch today may result in something that has bird-like characteristics (in the same way that you can kinda see the primate in humanity), but it won't be the parrot of day anymore than we're the chimpanzee of today. I don't entirely agree with @Nosajimiki that your birds would be required to evolve into humanoids... but I completely agree that they'd need to change to accommodate everything I've discussed.
Keep in mind, we don't have fur all over our bodies anymore because we've evolved to no longer need it. But we didn't need or use it for mating, either. What would a bird's plumage evolve into if it no longer needed it for warmth, etc.? And how would that hang around (kinda like human hair...) if a colorful display (not necessarily plumage!) was still needed to attract a mate?
To conclude... what you want to design is more important than the Real Life processes that might (maybe) lead to its design. Start with a little science... but don't be completely bound by it or you'll have to go where the proverbial evidence takes you. Which might very well be the flying cow @Vesper mentions. I liked the flying cow!