I don't understand how you can't understand this. Yes, I have a lot of issues with them. But they are definitely not a solution looking for a problem.
They supposedly have 128 m accounts in the US, and there are circa 150 m total people employed here. Even assuming account inflation, I'm sure the majority of professionals have accounts. Certainly when I'm screening resumes, it's very rare for a person not to be on LinkedIn. They've been going since 2002 and are a profitable business that sold for $26 bn. This should be a sign to you that even if it's not your thing, somebody's getting value.
Paper resumes were a giant pain in the ass. LinkedIn is a pretty obvious solution to the problems with paper resumes, including that they are hard to write, a pain to update, hard to format, impossible to search, and impossible to use in aggregate. This is obviously great for people hiring, but also great for those seeking employment. During a job search, LinkedIn lets you ask questions like, "Who do I know who works at company X? Who do I know who can introduce me to somebody at company X? Who used to work at Company X so I can get the unvarnished truth about what it's like to work there?" Try that with paper resumes.
TL;DR: People use it because it makes finding jobs and/or employees easier than what went before it. Lots of people.