The non-controversial part (that even Apple concedes in its briefs) is that the AWA grants courts the power to issue one of several particular forms of writs (court orders) that existed in British Common Law prior to the American Revolution, when it's necessary to do so in service of some other order that the same court has already issued.

The controversial part is what the AWA authorizes (if anything) beyond those pre-existing types of writs. Apple's position was "not much, if anything". The government's was, in effect, "anything not specifically denied by a law passed by Congress" -- which attracted heavy criticism, not only from Apple's lawyers, but also from many highly qualified lawyers who submitted amicus briefs.

I added that in an edit, but basically, it doesn't grant any powers by itself. It simply aids the court in carrying out its jurisdiction.

So, when it comes to wiretapping, there are specific statutes that authorize wiretapping in such-and-such circumstances. The court can then issue writs as needed to carry that out, provided the writs meet fairly narrow criteria (not overly burdensome, only targeted at an entity "close to" the case at hand, only targeted at an entity whose actions are "necessary" and where there's no other way to carry out the jurisdiction, rather than, say, just convenient) and the writ must have some kind of antecedent in common law, which is a somewhat fuzzy topic, but it boils down to "the writ can't just be pulled out of thin air, there's a set of traditional writs and it should appear to be consistent with that set".