For many, I think there was a pretty simple explanation: Halo. The hardware was a little iffy, but XBox has always had the mother of all console exclusives upon which to build their ecosystem. (Though the significance of the Halo franchise seems to have faded a bit in recent years.)
Halo is what sold me. Also, I was "in to computers", so the fact that Microsoft was making a video game console was pretty cool to me.
I was 11 years old when the first Halo game came out, and I got the original XBox/Halo for Christmas that same year.
Before that, I was playing Final Fight on Sega CD. So going from that to Halo was a real eye opener. I had never experienced emotion like that in a video game before. The music was EPIC, the story line was immersive, and the graphics were excellent for the time.
Halo is considered (by me) to be the first game of the new era. Sure, there were other successful "3D" FPS games that were out before Halo, but the original Halo hit all the marks and was the first taste of modern gaming for a lot of people. Many of my friends went from playing Mario/Sonic/etc... straight to Halo. It represented a massive leap in graphics quality, so much so that even games of today don't look _much_ better than Halo to me.
Halo is the only reason I purchased an XBox 360 and XBox One.
Yeah Halo was it for me, and probably the only real reason I own an Xbox One (although we'll just say that Halo 5 leaves something to be desired...) I think Microsoft has stumbled recently with console exclusives. A lot of people felt that Titanfall[1] was kind of a bait and switch. People paid $60 or whatever for what was essentially a multiplayer-only game. (The sequel will not be an Xbox exclusive.)
