You need some way to differentiate it from other forms of games. "Games" could mean board games, card games, sports, tabletop games, drinking games, party games, etc... One thing you should notice is that all of those (except sports) use the same naming syntax as video games IE descriptor + game(s). Video games are games played with a video aspect. This can then be further broken down with other terms such as console game, pc game, PS3 game, role playing game, etc.
We could start calling them computer games, I suppose, but even though that is technically correct (all consoles are computers) I suspect many console users would feel insulted/upset. Furthermore, use of said term could lead to confusion, though more limited than the generic word "game." Currently when someone says "computer game" most people think of PC/Mac games.
TLDR: When describing a game, terms are added to specify what you actually mean. "Game" could mean anything whereas "video game" means a certain subsection of games and "computer game" is an even more narrow subset.