Many kinds of considerations convince people. Everyone, not just philosophers, naturally sorts those considerations into "good" reasons and "bad" ones. People might sometimes disagree on where to draw the line, but most everyone agrees there's a line to be drawn. The good reasons are considerations that are relevant to the truth of the claim in question; the bad ones, irrelevant. Relevant in this sense: the truth of the considerations demands, or at least makes more likely, the truth of the claim being argued for.

Turns out that we've made a lot of progress in understanding this relation of relevance. Logic studies one corner of it: that which concerns entailment relations between claims, that is, when the truth of one proposition forces the truth of another.

With this knowledge in hand, we can see that people are often convinced by arguments that do not provide good reasons for their conclusion. And also, that they sometimes fail to be convinced by good arguments. Arguments that convince people and those that are good are not (sadly?) the same. It falls to psychology to study the kinds of arguments that convince people, because explanations for why people have the beliefs they do often make reference to features of their mental make-up.

Your last question, about why we should trust Logic, is a very interesting one. Does it help to realize that you have no choice? Not in the sense that you are psychologically compelled to trust Logic, because, as just mentioned, you aren't. But rather in the sense that the whole practice of asking whether some tool should be trusted is one that involves giving arguments, weighing evidence, in short, one that involves Logic. You might say that the laws of Logic characterize what counts as a justification, and so it's a bit odd to ask for a full-blown justification of Logic.