How does Google Scholar find papers on personal websites?
(Warning - gross oversimplifications incoming - if somebody doing research in Information Retrieval wants to add technical detail, be my guest!)

Fundamentally, Google finds all resources (HTML pages, images, as well as papers) on the web in the same way: periodically revisiting each and every resource it knows about, (re-)indexing it, and, for HTML content, following all links to other resources (rinse and repeat). Your web page is likely linked from your department web site, which Google definitely knows about, hence your web page is also in Google's database. Your web page links to your paper, hence Google will also know about your paper the next time the crawler checks your page. How long this will take is undefined, but Google has a lot of crawlers and is pretty smart about when to re-check certain types of pages, so it typically does not take very long.

Now, Google has specific heuristics in place to treat different types of resources differently. For instance, if an HTML page is added to the database keywords will be extracted, links will be followed, etc., while an image will lead to completely different actions. Scientific papers are not different in that sense - as soon as Google finds a PDF or Word file that "looks like" a scientific paper to the automated process, Google will generate paper metadata (title, authors, venue, keywords, ...) by parsing the PDF text as good as it can and add it to its special Google Scholar database, and this is when the paper appears in your profile.

Google's own website goes into quite some detail on this process. It also has instructions for authors looking to get their papers indexed by Scholar.