The creature is an ape-like quadruped very similar to humans. Their intelligence and feelings are the same as in regular humans. The head and torso are indistinguishable from a regular human. The arms and legs are the same length, and all four feet resemble regular human feet. Their neck is more flexible, allowing raise their head up and see where they are going. How could this creature evolve?
Although it isn't impossible for a humanoid to have quadrupedal movement, see lots of monkeys, the human form is very unfriendly for quadrupedal movement. The inefficiencies come from the way the arms and legs are build, of which their length is one of the limiting factors. It is simply not an efficient way to move. As the restrictions also tell of regular feet, we cannot use the joints there to adapt them for better locomotion. Although this would require major remodelling of the arms and legs. Not to mention the power transfer of feet to arms and the reverse will likely be a nightmare to deal with.
As it's so inefficient, the likelihood of this occurring is very slim.
It isn't impossible, though in practise you can say it is. To still get such humans, it seems easiest to start with humans and only transform their hands. This will take a lot of selective breeding, so either outside forces are doing this or we got the weirdest feet fetish all of a sudden. At the same time, we should only select the right size of arms and legs, to prevent any optimisation that nature would likely try to add. After many, many generations, you might end up with feet instead of hands, together with some difficult energy wasting quadruped movement.
There is a genetic form of anemia called sickle cell disease. It makes you generally less healthy than people that don't have it, so the genes for it tend to be fairly uncommon in most of the world. However, in some populations of Africa, it helps you survive:
The malaria parasite has a complex lifecycle and spends part of it in red blood cells. In a carrier, the presence of the malaria parasite causes the red blood cells with defective haemoglobin to rupture prematurely, making the Plasmodium parasite unable to reproduce. Further, the polymerization of Hb affects the ability of the parasite to digest Hb in the first place. Therefore, in areas where malaria is a problem, people's chances of survival actually increase if they carry sickle cell trait (selection for the heterozygote).
Therefore it is actually "good" to have that condition.
Just the same, some people walk on all fours due to a cerebellar ataxia. There is even a documentary about a family in which the condition affected quite a few members:
First, their mother recalls that initially all of her 19 children started off walking with a bear-crawl (i.e. on their feet rather than their knees). Second, due to an inherited recessive genetic mutation, they have a non-progressive congenital cerebellar ataxia that impairs the balance children normally use to learn to walk bipedally. Not being able to manage the balance needed for bipedal walking, they perfected in its place their initial bear-crawl into an adult quadruped gait.
You need to have a reason why walking on all fours would be beneficial (maybe this helps populations that live in caves with a low ceiling - it would make it easier to build and live in cave cities such as Derinkuyu). Over millennia or even millions of years new genes would come up to modify the hands and general body plan into a pronograde one.
Not Hoffmann or Huysmans could conceive a scene more wildly incredible, more frenetically repellent, or more Gothically grotesque than the twilit grotto through which we seven staggered; each stumbling on revelation after revelation, and trying to keep for the nonce from thinking of the events which must have taken place there three hundred years, or a thousand, or two thousand, or ten thousand years ago. It was the antechamber of hell, and poor Thornton fainted again when Trask told him that some of the skeleton things must have descended as quadrupeds through the last twenty or more generations. Horror piled on horror as we began to interpret the architectural remains. The quadruped things—with their occasional recruits from the biped class—had been kept in stone pens, out of which they must have broken in their last delirium of hunger or rat-fear. There had been great herds of them, evidently fattened on the coarse vegetables whose remains could be found as a sort of poisonous ensilage at the bottom of huge stone bins older than Rome. I knew now why my ancestors had had such excessive gardens—would to heaven I could forget! The purpose of the herds I did not have to ask...
Your quadrupeds are the descendants of human livestock, kept as meat animals over thousands of years. The historic fondness of the keepers for quadrupedal humans is just one of their many unusual fondnesses.
It's not impossible but very unlikely. Being bipedal gives us an enormous evolutionary advantage over quadripeds - we can use our hands to make stuff. For humans to evolve into quadripeds again the selective pressures on humans would need to shift away from tool making and towards flight or pursuit but I can't see how that would happen.
Human feet evolved for an upright posture. An animal with four humanlike knees and four humanlike feet simply does not make evolutionary sense. Consider the lack of existing quadrupeds that have anything like a human foot with its ankle joint and additional joints at the toes.
If the environment has a lot of slippy surfaces, a bipedal gait might be unpractical.
One way could be moss-covered stone, which can be extremely slippy (also pairs well with "The Square-Cube Law"'s idea of a cave civilization). Consequently, keeping a low center-of-mass reduces injuries from falls and the additional "actively walking limbs" make it easier to keep traction.
Alternatively, a large portion of the world could consist of frozen lakes. In addition to the points above, distributing the weight over more limbs reduces the risk of breaking into the ice and drowning.
Instead of starting with existing humans, start with the quadruped great ape humans evolved from (See Hominina). All you need to do is to add more brain size. Typically, larger brains come from hunter species. So, make it a hunter.
That will give you a different body structure than you specify, but it would be the easiest evolutionary path. The body needs to be evolved for quadrupedal action which means a different torso shape and the hands would be different - more like chimp hands. The human face comes from eating cooked (softer) foods. That means that we don't need the muscles and teeth needed to deal with fibrous plants or canines to tear off meat. Figure out how a quadruped would start using fire and a more human face can appear.
Humans have two major mutations from the great apes: bipedal walking, and speech.