How does Blistering Dieflyn work?
First off, though most often you get your mana from lands, it's not "when you use a swamp or a mountain" but "when you pay a black or red mana".

And yes, you can activate the ability multiple times, for example paying 2 red and 2 black mana to give it a total of +4/+0.

This is called an activated ability, as is anything else of the form "[cost]: [effect]" (note the colon). You can activate an activated ability as many times as you want, as long as you can pay the cost. Sometimes their cost is tapping (for example Cinderhaze Wretch), in which case since you can't tap something that's already tapped, you'll only be able to use the ability once at a time. But in this case, the cost is simply mana, which you can pay repeatedly.

Note that you are technically activating the ability multiple separate times, not just paying more for a bigger effect all at once. The difference doesn't usually matter, but in some cases it can be handy to do something in between activations.