Now I understand that Microsoft has a competing technology in the web server field with the Internet Information Services, so not every last web server is done with Linux, but how did Linux web servers become so ingrained in the web server sphere? Does the fact that it is an open source solution have anything to do with it?
Unless someone can come up with some actual market research from the 1990s ("we've asked n companies which web servers they chose, and why"), this will be opinion based, but:
I'd venture that (a) Unices had a head start as internet servers, and that (b) Linux was the cheapest good-enough platform.
Regarding (a), there were many internet services (mail, Usenet news, FTP, Gopher) that predated the web. Their servers mostly ran on Unix or VMS machines, and VMS' star was fading by the early 90s. For what it's worth, the first webserver, CERN httpd, first ran under NeXTStep, a BSD-derived Unix(oid Operating System).
Regarding (b), keep in mind that in the 90s, there were several commercial Unices: SunOS / Solaris, IRIX, HP-UX, AIX, and I'm probably forgetting several; and there were several BSD variants. But Linux ran on commodity hardware and was free.