Whenever you see a word that starts with "ex-," that prefix "ex-" usually means "from, off, out, or very;" and in the word "exasperate," it means "very." The "asperate" part means "make rough." And so, to exasperate someone is to make them act very rough, usually by bothering and annoying them until they blow up.
In other words, when something exasperates you, it annoys you a lot. Exasperating things either take too long or happen in a way you don't like. And when things exasperate you, you're so annoyed that you stop acting kind and gentle and you start acting rough and mean in response.
For example, if your little sister repeats everything you say, long after you ask her nicely to stop, she's exasperating you. You might yell at your sister, which isn't nice at all—but you ran out of patience. You were exasperated.
For another example, if you're playing a video game and it keeps crashing and shutting down on you, again and again, then it's exasperating you. And you're exasperated! You might shout "This game is stupid!" and slam your fist onto your desk, which is a very rough way to behave: you're exasperated.
Exasperation is a feeling we all experience now and then. Even though things and people exasperate us, we're still in control of our responses, and we can try hard to keep calm and stay kind. What do you do to ease your sense of exasperation, to stop yourself from yelling or from hurting things (or people)? When I'm feeling exasperation, I take in deep breaths and count to ten very slowly.