Looking deeper into FanGraphs' glossary, I got the formula to how they calculate this stat:
Clutch = (WPA / pLI) – WPA/LI
Avila had a WPA of 1.02, a pLI of 0.89 (1.02/0.89 = 1.15) and a WPA/LI of 0.95 (1.15  0.95 = 2.10). There's probably a rounding issue here.
So what do these stats mean? WPA is Win Probability Add. It's a measure of the change in win expectancy from the time a batter enters that batter's box until the end of play. Without getting too technical, there's a win expectancy chart that shows the percentage of how likely a team is going to win the game at any particular moment in the game. For example, say a team has a 45% chance of winning a game. Then a batter hits a HR. Now the team has a 75% chance of winning. The difference (0.75  0.45 = +0.30) gets credited to the player. You may have already seen this in chart form:
Source: FanGraphs
pLI is the player's Leverage Index in all situations. It attempts to quantify the "pressure" aspect of the game by factoring in the inning, the score, the number of outs and the number of runners on base. Think of bases loaded in the 9th inning of a tie game as a really high leverage situation and a 130 blowout in the 5th inning a really low leverage situation. Neutral is given a 1, "high leverage" is considered to be 1.5 and above and "low leverage" is below 1. Avila's 0.89 is right around 1, which is to be expected.
WPA/LI is context neutral wins and is calculated on an individual play basis and then added up during the season. During the game, some players will see more at bats in higher leverage situations than others, so this stat puts everything on a level playing field. Sometimes a walk is just as valuable as a HR, so this stat will show that.
The subtraction of these two will result in showing only high leverage situations. So what we are seeing is how much better a player performed in high leverage situations compared to how he performed in low and medium leverage situations. FanGraphs also gives splits on these situations, so we can see them in more traditional stats:
Source: FanGraphs
pLI is the player's Leverage Index in all situations. It attempts to quantify the "pressure" aspect of the game by factoring in the inning, the score, the number of outs and the number of runners on base. Think of bases loaded in the 9th inning of a tie game as a really high leverage situation and a 130 blowout in the 5th inning a really low leverage situation. Neutral is given a 1, "high leverage" is considered to be 1.5 and above and "low leverage" is below 1. Avila's 0.89 is right around 1, which is to be expected.
WPA/LI is context neutral wins and is calculated on an individual play basis and then added up during the season. During the game, some players will see more at bats in higher leverage situations than others, so this stat puts everything on a level playing field. Sometimes a walk is just as valuable as a HR, so this stat will show that.
The subtraction of these two will result in showing only high leverage situations. So what we are seeing is how much better a player performed in high leverage situations compared to how he performed in low and medium leverage situations. FanGraphs also gives splits on these situations, so we can see them in more traditional stats:
2013

AVG  OBP  SLG  wRC+ 

Low Leverage

0.218

0.317

0.324

80

Medium Leverage

0.225

0.316

0.383

93

High Leverage

0.290

0.324

0.645

148

And it's clearly noticable that Avila performed much better in high leverage situations than in low and medium leverage situations. The difference between them was better than the difference of any other catcher last year. And better than any other Tiger batter last year. In fact, Avila was the most clutch player of anyone with at least 300 plate appearances last year.
One last thing to keep in mind  there is no predictability value here. Just because a batter preformed well in high leverage situations one year, doesn't mean he'll continue to perform that way every year. This is why several people claim there is no such thing as a clutch hitter. To prove this, look at how Avila has performed in high leverage situations in the past:
High Leverage  AVG  OBP  SLG  wRC+ 

2011

0.224

0.333

0.408

85

2012

0.295

0.360

0.477

127

2013

0.290

0.324

0.645

148

While Avila performed pretty well in High Leverage situations in 2012, in 2011 he was pretty poor. If you remember, 2011 was Avila's big breakout year (.295/.389/.506, 140 wRC+ overall).
FanGraphs says that this is a good storytelling stat. And that's all this is, a good story.