Go is hard for computers primarily because the board is huuuuuuge. Do Go on an 8x8 board and I suspect computers would just crush humans.
Go is pretty much never played on 8x8, but it is sometimes played on 9x9 by beginners, people looking for a fast game, and people who want to just have one tactical battle only. Computers are indeed very good at 9x9 already, as in professional level (= grandmaster level in Chess).
Well generally for a game like Go or Chess you're not going to compute every possible remaining game state (that would take way too long), but you can use a number of different strategies to make good moves. Like, you could look a few moves into the future and choose the move that results in the best game state for you assuming your opponent plays optimally (minimax), or do a genetic algorithm where you "train" a computer with various sample games (this is what they did with IBM's Watson to train a computer to play Jeopardy).
The newer Go programs use a "monte carlo tree search" which involves taking candidate moves and playing out the rest of the game randomly (with some pruning I'm assuming) many, many, times. The program then chooses the move that maximizes the probability of winning. This seems to work a lot better than brute force minimax searching. It's probably also what leads to a lot of the really strange-looking moves (for example, if the computer thinks it is ahead, it will play an uber safe move that results in it winning by 0.5 points rather than something that a player of its level could easily read out as leading to a 20 point victory).
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your 7 time champion, Link.