There is nothing saying they cant shake hands, generally speaking unlike in other sports, only 1 team is on the playing area when the game ends. The winning team usually takes to the field to "celebrate" or shake hands with one-another. Also in Baseball unlike in other sports you play the same team 3-4 days in a row and will play some teams in your division 18-19 times in a season.
Although baseball players don't take the field to shake hands after a game or series, you will sometimes see a waive or a 'tip of the cap' coming from one dug out to the other, this gesture is the equivalent to a handshake in other sports
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people shake hands because it's like saying good game but without talking and in a nicer way.
Because they want to show eachother that its not going to be a friendly game
Hope that helped