It was easier to learn than the alternatives (UDK, any of the Java engines) and consistently added clearly useful features.

Jump to 2019, their networking solution, UNet is abandoned. Multiplayer is a perhaps a top feature developers/players want. A great open source library called Mirror picks up the slack, but Mirror wasn't made by Unity.

As other commenters have mentioned, easily compiling to iOS made it a clear winning choice for game app development.