PDF was created to solve the problem of being able to render a document the same way on different computers, and it mostly achieved that goal. Editable formats like .doc, .html, .rtf were unreliable -- different software would produce different results, and even if two computers have the exact same version of Microsoft Word, they might render differently because they have different fonts available. PDFs embed the fonts needed for the document, and specify exactly where each character goes, so they're fully self-contained.

After Acrobat Reader became free with version 2 in 1994, everybody with a computer ended up downloading it after running across a PDF they needed to view. As it became more common for people to be able to view PDFs, it became more convenient to produce PDFs when you needed everybody to be able to view your document consistently. Eventually, the ability to produce PDFs became free (with e.g. Office 2007 or Mac OS X's ability to print to PDF), which cemented PDF's popularity.

Notably, the original goals of PDF had nothing to do with being able to copy text out of them -- the goal was simply to produce a perfect reproduction of the document on screen/paper. That wasn't enough of an inconvenience to prevent PDF from becoming popular. (Some people saw the inability for people to easily copy text from them as a benefit -- basically a weak form of text DRM.)