Honestly, it is difficult to put into words just what it is that made GTA so entertaining to me (or why Vice City utterly failed to match GTA3).
Shenmue was one of the very first games to really excite me about a virtual world. Just exploring the world and seeing the sights was enough to interest me. While I ended up enjoying Shenmue in the end, it failed to deliver what I was hoping for; a large, open world with no visible loading. The missions were great fun and well designed (for the most part), but outside of those missions, I could simply enjoy cruising the world. Taking a walk down the street and obeying the laws while admiring my surroundings was someone a serene experience. It was the initial discovery of everything that really impressed me.
The voiceless character you played in GTA3 combined with the less obtrusive comments coming from others on the street really worked well. Watching the sun go down by the ocean, walking along a deserted river bank as rain poured down in the middle of the night, finding ways to climb various buildings and structures and looking out over what I just conquered, etc. Those kinds of activities were entertaining and worked great within the context of the missions.
I enjoyed GTA3 for the same reasons I would later enjoy Morrowind (Daggerfall came before both, of course, but it was simply too limited by technology to really compare). You have a game that provides a large number of fun, variety filled missions that allow a lot of freedom in their approach backed by an exciting and interesting world (though, Morrowind's world was quite stagnant...but there was even more depth to be found, sooo...).
Sure, it's violent at times...but a mature, stable person is not going to take what they see and apply it to the real world. Unlike movies, GTA doesn't really look a whole lot like reality. Fictional violence does not bother me one bit. Though, there are exceptions. Irreversible (a movie), for example. -shudder-