What evidence is there to say that the dinosaurs were destroyed by an asteroid

I do not think that anyone has made the claim that this was the sole reason for their extinction. There were probably several factors, including climate, disease, and competition for food. However, when Dr Luis Alvarez and his son discovered a relatively thin layer of iridium while researching the interface between the soils that were deposited during the end of the dinosaur age and the time immediately afterward, they provided a key piece of evidence. This was a layer of iridium dust that was clearly deposited right when the dinosaur extinction occurred. The levels below it contained dinosaur fossils; the layers above it contained only mammals, birds, amphibians, etc - but no dinosaurs. Now, the interesting thing about iridium is that it is pretty rare on earth, but quite common in meteorites. And the other interesting thing is that this same discovery - in the same time frame - was subsequently repeated all over the world in over 50 different locations. Finally, the depth and thickness of the iridium layer varied across the globe - but was most pronounced in the regions surrounding a known large meteor impact site in the shallow waters off the Yucatan Peninsula that occurred at roughly the time of the dinosaur extinctions.

When you look at all the evidence, it points to a likely significant meteor impact that covered the globe with iridium dust about 65 million years ago. Before the impact, there were dinosaurs. After the impact, there were few or none. When the data adds up like that, it is prudent to listen to what it is trying to tell us.

Dinosaur fossils occur up to the iridium layer marking the time of impact of the asteroid, but no non-avian dinosaur fossils have ever been found above the iridium layer (there is one possible exception in West Texas).

Calculations show that the asteroid was large enough to have caused worldwide effects major enough to account for the extinction.

Many other forms of life, especially terrestrial forms, went extinct at the same time as the dinosaurs, indicating that the cause was not something like a disease that would have affected only dinosaurs and perhaps their relatives.

The ones I know of: Mass extinction at the same time, deposits of iridium of the same age all over the world, suggesting a debris from a large extraterrestrial object at that time, and a huge crater of the right size and age in one of the oceans.

There are other pieces to the puzzle too, but those are the ones I've read of.