Why are journals used in modern scientific academic research?

Why are so many people publishing [on ArXiv]?

You have to be careful with terminology when making statements like that. ArXiv is certainly "publishing" in the literal sense of "making public" but would you say that you'd "published a book" if you'd just put it on your website? Probably not.

My question is, why were journals used to begin with?

Because they predate ArXiv and the web by hundreds of years, though the format has changed over time.

Why are they used now?

Partly through inertia, partly through a lack of alternatives. On a purely scientific level, ArXiv itself is not peer-reviewed. On a non-scientific level, ... People have an intuition about the quality they expect from a paper that appears in a particular journal and we don't have alternatives that allow us to make this kind of judgement. Correspondingly, certain journals have quite a lot of prestige, which is important for students early-career researchers who are looking for promotion; a large fraction of papers have at least one such author. It's easier for people like Tim Gowers to work outside that system.

First of all, ArXiv covers mathematical disciplines (specifically: Physics, Mathematics, Computer Science, Quantitative Biology, Quantitative Finance and Statistics), not all academic fields. So, the question, which seems to pose this as a general inquiry about the need for journals, is overly broad in the context of offer ArXiv as an alternative.

Second of all, ArXiv is not peer reviewed. Therefore, consumers of the information need to be a bit more careful than when they're referencing information that has gotten past an editorial board and at least a couple peer reviewers. That's not to say that erroneous material doesn't get past peer reviewers (it does) but there's just more quality control in a peer reviewed paper than an unreviewed paper.

Third, my impression of ArXiv is that people upload their stuff there to make it available sooner while they pursue publication in a peer-reviewed journal. It is not meant as a publishing destination, but rather a repository to speed up dissemination of the information for those who may want it sooner.

Fourth, in my department (I'm in a science discipline, not math), getting tenure requires publishing in peer reviewed journals that are indexed by something like Pubmed. Book chapters and unpublished reports (which is what I'd consider an ArXiv paper) count very little toward fulfilling tenure expectations.

Peer review, as the others have said.

Mathematicians can consult dozens (if not hundreds) of false proofs of the Riemann Hypothesis on Arxiv. Such things are (mostly) rejected by the journals. Journals which publish them soon get a bad reputation.

Imagine if all the amazing new health claims form the Internet could not be distinguished from serious medical research! (For many people I know, that is unfortunately already the case. But those who care can try to find which of those cancer cures have at least some actual evidence for them.)