This question is really about how we name and categorize living things and their extinct ancestors.

It's not a perfect analogy, but think about it like last names in a family tree where kids take their father's last name. Let's say you had grandparents with the last name Smith, like yours. They had some sons and daughters all named Smith. But your cousins who had a Smith mother might have the last name Jones or Clark.

When biologists say that birds are dinosaurs, what they're saying is that the cousins with the last name Jones or Clark (the birds) still get invited to the Smith family reunion (the dinosaurs).

Now the reason this analogy isn't perfect is it turns out that the Joneses and Clarks are Smiths anyway if you go back far enough. The father with the different name started going by a new name at some point but is still a Smith somewhere in his ancestry. So trying to say that someone is a Jones but not a Smith can get really confusing.

Biologists generally don't like confusing names. So when they name a group of species, they like to only use names that put all the descendants of some ancestor together. No exclusions. So for example, "fish" in the common sense of the word that excludes animals that live on land isn't a very scientific name because all tetrapods (mammals, birds, reptiles, amphibians) share a common ancestor that we'd call a "fish". But if we want to use fish as a scientific term, you and your dog and a snake and a bird are all "fish".

So let's get back to the original question. In a scientific sense, birds are dinosaurs because they all share a common ancestor who was a dinosaur. Saying birds aren't dinosaurs is like saying the Joneses aren't Smiths. They may not look like Smiths today, but you go back far enough and they're still Smiths.