"y los movemos continuamente para que no se nos quemen y se vayan tostando." Means: and we stir them continuously so that they don't burn and just keep toasting.

They don't burn that quickly, it's just common sense to keep stirring so that they brown evenly. Much simpler than toasting sesame seeds or cashew nuts in a pan. I'd suggest 1 tablespoon of oil for each 100g of pasta.
Also, don't worry if a few of them do burn a little. They still taste good. You'd have to be really inattentive when cooking them to burn them enough to ruin your dish.

As well as changing colour you'll see that the strands actually puff up a little.

Even though it might not be "the norm" it is pretty common to fry pasta and noodles before boiling them, the techniques vary so sometimes freshly made pasta is fried (like Cantonese Yi Fu Noodles) or dried noodles might be par-boiled and then flash fried before packing (instant ramen and other Eastern made instant noodles are a common example). I'd guess that most Spanish fideos aren't pre-toasted and they are just added to chicken stock. Dried rice is often fried before boiling also, may not be the most common way to prepare rice, but it's a worldwide practice all the same.