How did scientists know that Dinosaurs became extinct and didn't just evolve into something else?

Dinosaurs became extinct AND evolved into something else. When a species no longer exists it is considered extinct, but if a species manages to live through thousands of generations, there is a good chance that at some points certain offspring will have mutations that make it more successful in its environment. Enough mutations and successful offspring creates a new species. The modern day animals that most like likely evolved from dinosaurs are modern day birds.

http://en.wikipedia.org/wiki/Origin_of_birds

EDIT: And as note to another answer, ancestors of crocodiles did exist during the time of dinosaurs, but crocodiles did not evolve from dinosaurs, they are reptiles. It is believed that most dinosaurs were warm blooded.

Exactly. Fossil evidence can be carbon-dated, and there are no 'modern' dinosaur fossils. We know that something cataclysmic must have happened to their environment to have them disappear so quickly, and many paleontologists and geologists feel that it was a huge collision of an enormous meteor, or space debris, that may have created the Gulf of Mexico. A strike that big would have raised clouds of material and water vapor into the atmosphere which would have blocked the sun and affected the living conditions on Earth for many, many years and may have caused such a rapid change that the dinosaurs could not cope.

There is more and more evidence to support that some of the dinosaurs, such as pterodactyls, did indeed develop by natural selection over many generations to form modern birds, which also supports the growing theory that dinosaurs may, indeed, have been warm-blooded and not the cold-blooded reptiles we once thought they were.

Descendants of dinosaurs still roam Earth. Crocodiles are the closest still living relatives of dinosaurs.

Most of the extinct species show no fossil record beyond the time Chicxulub crater formed, which led to the hypothesis that that was the mass-extinction event that killed most of the dinosaurs.

They went extinct either way. We know this because they aren't here. Scientists believe that many of the old species did evolve into new ones. Most of them died out completely.

Extinct just means that the species died out. It really doesn't matter how.

Some scientists specifically say that. Namely, that Dinosaurs (some) evolved into birds.

Some of the smaller, feathered dinosaurs did survive and evolve into something else: our modern birds.

As far as most other dinosaurs, I think it's because we don't see fossil evidence for their existence post-Cretaceous period.

Because the fossil record of them ceases abruptly. If large dinosaurs slowly evolved into other forms, we would see evidence of such change in the fossil record.

Because before the K-T boundary layer, there are large dinosaur fossils; and after, there aren't.

Scientists don't actually know any such thing, but they will never admit it.