Let's see, I had some exposure to DT from various forms of media in the last 15 years. First time would probably have been when I watched the Dragon Ball Z tv film: The History of Trunks some time around 2001. I actually remember thinking how awesome the soundtrack was for that movie but I had no idea who it was and I wouldn't know for another decade or so that is was a mostly DT soundtrack (SFAM instrumentation specifically).
About 2004 a friend of mine from high school (he was a drummer, I played some guitar and keyboard so we jammed occasionally). One day he showed me a track from this band he quite liked; it was Honor Thy Father and the drum fill at the start blew his mind and he told me about this Mike Portnoy character and how he was inspired to try play like him. The whole song was so intense and full of crazy riffage and was about 10 minutes, I'd never really heard anything quite like it. Most metal I'd heard was fairly generic and simple. So at this point I knew who DT were but it wasn't enough to get me searching for more.. I have know idea why, probably laziness and being content with most of the music I was listening to at the time.
2007 I played some of those guitar hero and rock band games. I enjoyed the rhythm gameplay and actually got quite into it for a while, being a keyboardist I was able to pick it up fairly quickly (funnily enough, I'd compare those fake guitar games more to playing a keyboard than to playing a guitar as far as finger technique and movement goes because of the obvious buttons instead of strings). So we have Pull Me Under on one of the games, Constant Motion as DLC for another and then Panic Attack on yet another game. I loved all 3 of these songs because they were epic, long and rather challenging. (CM was considered to have one of the hardest guitar solos in the game around when it came out) and I was able to recognise the name
Dream Theater at this point. I can't believe these songs weren't enough to get me digging into the bands material though; for shame...
Then 2009. My windows media player tries to advertise something to me, it was a preview of A Rite of Passage. I really liked how different yet how catchy it was. And something finally clicked... I went and got BL&SL nearly straight away and started going backwards from there and a few months later I had ALL of the DT material I could get my hands on and was becoming quite familiar with their discography. And have been following them as one of my favorite bands ever since.
So yeah, that about briefly some it up. By the way, that's the short version of the story.
Interestingly, On the Backs of Angels came out on the Rock Band Network in 2011 some time, I kinda stopped playing those games like everyone else because they over-saturated the market, but RB3 actually uses real instruments including an actual 6 string guitar and a 2 octave keyboard. I would be really curious to try play the keyboard parts to OTBOA on this version of the game but I'd have to justify spending a few hundred dollars on the new instruments that may be of questionable quality. Funny how those games were just starting to get better (more authentic); right about when people were getting sick of them. There is that whole rocksmith thing now too which looks curious but I haven't been able to check it out yet.