A definition:

RE24 (runs above average by the 24 base/out states): RE24 is the difference in run expectancy (RE) between the start of the play and the end of the play. That difference is then credited/debited to the batter and the pitcher. Over the course of the season, each players’ RE24 for individual plays is added up to get his season total RE24.

the stat tells you what the hitter has done to increase or decrease the likely runs as a result of the plate appearance. It does not tell you how much those runs added to the likelihood of team victory.  That is WPA.  Intuitively, a single with men on first and third and no outs in a tight game adds more to the likelihood of a team winning than a single with the same base runner places and outs in a blowout, but both singles add the same run expectancy. 

One difficulty in using it to compare players would be that it is a counting stat. A guy who has more opportunities and who is above average is going to get a higher RE24 than a guy who is just as good (or maybe better) but gets fewer opportunities.  This might be the case comparing Adam LaRoche to Tyler Moore, for example.  Batting order too might make a difference.  Desmond as a leadoff hitter would be less likely to pile up a big RE24 than Desmond hitting behind some good on base guys, batting 5th or 6th.   This is why a lot of people who were down on him hitting leadoff thought his contact-heavy low OBP would be better used lower in the order even before he had the big jump in SLG after April.  With the jump in SLG, it is more obvious.

I found this article from Tom Tango / Tangotiger in Fangraphs, pulled from The Book, which points out that this is might be a good stat to judge relievers.  It captures the difficulty and importance of mid-inning stoppers who come in with guys on and prevent runs from scoring. That is, the rescue squad that comes in after an H-Rod appearance