Well, evolution is a very complex thing. One minute an animal can be the perfect fit for an environment, and the next minute the environment can work directly against them. Dinosaurs were very, very successful animals right up until the asteroid impact. Dinosaurs were not in decline before the KPG extinction (which is a common myth).
The asteroid impact itself was extremely devastating and most dinosaurs probably died soon after the impact. However, if any survived, they were not in a very good situation. Plant ecosystems were devastated, so herbivores died from lack of food, and carnivores died because they couldn't get enough meat, because of the lack of herbivores. In fact, no tetrapods (four-legged animals) weighing over 55 pounds survived, except for a few cold-blooded animals like crocodilians and sea turtles.
Cold-blooded animals require much less food to survive than warm-blooded animals. So crocodilians had a unique advantage. They can go over a year without food, and they can tough out long periods of scarcity by staying really inactive. Dinosaurs didn't have this advantage. They were much closer to being warm-blooded, and they would've needed a huge amount of food to survive.
Even though some small animals survived the KPG impact, most of them actually died. Many types of birds, such as the toothed birds, died out. Lots of mammals died out. And of course, all the small dinosaurs died out. Animals that survived were usually animals that had a place of shelter. Lizards and other little reptiles could shelter away and tough out the long periods of food scarcity. Little mammals could burrow and escape the harsh climate, and they probably had adaptable diets and could feed on whatever was available. The same goes for birds, which probably also had adaptable diets and could search far distances for food. Small dinosaurs, however, probably didn't occupy the "burrowing & sheltering" niches.
Think about it this way: birds are just a lineage of dinosaurs that evolved to fill a niche that involved sheltering. Therefore, they were the dinosaurs that survived. The other dinosaurs didn't have this niche, so they died. It's pretty simple, but it requires looking at birds as dinosaurs, for it to make more sense.
There probably weren't that many dinosaurs small enough to survive the impact in the first place. The ones that did probably didn't last long. They didn't burrow or live in shelters, so they were exposed to the harsh climate. Food was scarce, and the generalist mammals & birds outcompeted them by having a broader diet.
In extinction events, animals often outcompete one another due to very slight advantages and disadvantages. Dinosaurs didn't die out because they were inefficient or anything - they just weren't really suited to the changes. Small dinosaurs died out alongside many types of mammals & birds. The mammals & birds that survived had a slight advantage over the other animals, due to their uniquely adaptable lifestyles, and that advantage was enough to get them through an extremely harsh time in earth's history, and come out on the other side.
Hope that explains it. For any paleo-experts out there, please correct me if I got any small details wrong.