What made them important: Eye of the Beholder is a classic and served as my introduction to Dungeons & Dragons on the computer, but I don't think the game has any progeny on the shelves today. I think a first-person, party-oriented, turn-based RPG would be an impossible pitch in today's market. (You could claim that something like Dragon Age is a descendant, but I think that would be a stretch.)
But what solidified Westwood's place in history is the fact that they basically invented the Real-Time Strategy genre. Bits of the idea had surfaced in earlier games, but Westwood's Dune II gave us the full package: Build a base, harvest resources, make dudes, and kill the enemy base. (Yes, Dune II - they invented a new genre in the process of making a sequel. Man, do I miss those days.) After Dune they went on to create the Command & Conquer games.
What happened to them: They were purchased by Electronic Arts in 1998, and closed by EA in 2003. The silver lining is that EA admitted their mistake and CEO John Riccitiello took responsibility for the once-successful company flaming out under his management. They seem to have mended their ways, and BioWare has fared much better in their service since then. Hm, admitting fault and changing direction? I disagree with Riccitiello from time to time, but this is a pretty good example of why he's a class act compared to other guys I could mention.
The number of treasured classics that EA is sitting on is just... wrong. The Eye of the Beholder series is one of many that could really use a GOG re-release.