I will use a specific example, but I mean in general. I went to a number theory conference and I saw one thing that surprised me: Nearly half the talks began with "Assuming the generalized Riemann Hypothesis..." Almost always, the crux of their argument depended on this conjecture.

Why would mathematicians perform research assuming a conjecture? By definition, it is not known to be true yet. In the off-chance that it turns out to be false, wouldn't all of the papers that assumed the conjecture be invalidated? I may be answering my own question, but I speculate that:

There is such strong evidence in support of the particular conjecture (Riemann Hypothesis in particular) and lack of evidence against it, that it is "safe" to assume it.

It's not so much about result obtained, but the methods and techniques used to prove it. Perhaps by assuming the conjecture, in the case of the Riemann Hypothesis, it leads to development of new techniques in analytic number theory.