Dino-birds or "Pigeon-o-saurus"
First, let me link a variety of popular science articles which can give you some background explanation. Here are Scientific American, Smithsonian, and Audubon magazines. 

The 'smoking gun' evidence linking birds and dinosaurs is the evolution of feathers. Feathers are a feature that set the birds apart morphologically from the rest of the tetrapods, and indeed from the rest of all animals. All birds have feathers and nothing that isn't a bird has feathers. So a logical place to start looking for the origin of birds is the origin of feathers. 

Since the dawn of modern paleontology, there had always suspected links between dinosaurs and birds due to the fossils of Archaeopteryx found in the 1860s. Thomas Huxley claimed the two groups were related as early as 1863. But it wasn't until recent decades that the fossil record started being filled in with definitive links between Archaeopteryx and therapod dinosaurs, such as more species intermediate between birds and therapods and fossil impressions indicating that some therapods had feathers. 

The rest is just filled in by logical deduction, or Occam's Razor. There is clear evidence that all birds have feathers, and no other animals alive today have feathers. There is now clear evidence that therapods had very similar skeletons to birds with feathers about 150 million years ago. There is also evidence that many therapods had feathers, a trait otherwise confined to birds.

Let our null hypothesis be that the origin of birds is unknown, and our alternative hypothesis be that birds evolved from dinosaurs. Given the similarities noted above (and, keep in mind, the evidence is much stronger for scientists who specialize in this area), do you think there is enough evidence to reject the null hypothesis and conclude that birds evolved from dinosaurs? If not, are there any plausible alternative hypotheses?

The great thing about science is that you never have to take someone else's word for it. Everyone has the right to be a skeptic, just keep an open mind and look at the evidence and you can draw your own conclusions.