WPA quantifies the percent change in a team's chances of winning from one event to the next. It does so by measuring the importance of a given plate appearance in the context of the game. For instance: a homer in a one-run game is worth more than a homer in a blowout.
As an example: When Josh Donaldson came to the plate in the bottom of the ninth on May 26, 2015, the Blue Jays trailed by two and had men on second and third with no one out. That gave them a 43-percent win expectancy. After Donaldson's walk-off homer, their win expectancy jumped to 100 percent. Because Donaldson boosted the Blue Jays' chances of winning by 57 percent, his WPA for that plate appearance was 0.57.
A player's WPA can also be affected on the basepaths. It will increase if he steals a base but decrease if he is caught stealing or picked off.
(Team's win expectancy after a plate appearance or SB/CS/PK) - (team's win expectancy before that plate appearance or SB/CS/PK).
Why it's useful
WPA should not be used as an indicator of future performance. But WPA is a fantastic "story stat" -- meaning it does a good job of putting context to what has already happened. Its best use is for deciphering the impact of a specific player or play on a game's outcome.
There are much better ways to better judge a player's skillset than by using his WPA. It's a metric based largely on situation, and it generally shouldn't be used for fantasy purposes.