Is it a meaningful statistic? Sure. It indicates the proportion of times a batter reaches base safely not due to an error when a baserunner occupies second or third base (or both) when he is at bat. That tends to mean that batter has been responsible for batting in a lot of runs (as a proportion of opportunities). Personally, I would group it in with Runs Scored, Pitcher Wins, RBIs, and other statistics that are very helpful for indicating how well a player did, but not necessarily indication of likely future performance (except insomuch as they may be proxies for what kind of team he's on).
Is it meaningful for projecting future performance? I doubt it; according to Beyond the Box Score, there is a .21 correlation year-over-year in team BARISP. They misuse the word 'significant' there, but it's forgivable as it's probably not meant as a statistics term. It is, certainly, a weak correlation; but it's not a zero one, so perhaps it has some small predictive value.
Thinking about it from a 'how' sense, you could attribute it to 'playing better under pressure', which is mostly nonsense but not entirely. You could attribute it to a particular batting style that favors contact and line drives over fly balls - I doubt Adam Dunn's BARISP is particularly good, for example (it's 20 points lower for his career than his batting average, in fact), but Tony Gwynn does have a higher BARISP than his career batting average, by 14 points. Maybe some players change their swing in ways that happen to yield a better average - with a guy on second you swing for a single, but with the bases empty you just go for the home run (again, see Dunn, Adam for a great example).
Thus, it seems to me that it's not a 100% pure junk statistic; but with a low correlation coefficient, it mostly is, and particularly as it's used for a "clutch" indicator (which most statisticians laugh at). More likely it explains for certain players intentional changes in their game geared towards scoring runs, and for most players just signifies the noise that is random number generators, aka baseball at-bats. (My next pseudo-random number generator will be based on an initial seed, and the outcome of each at-bat in a major league season. Why not?)
Finally, I'd warn of the risk of circular reasoning. We often see 'good' batters singled out for having a good BARISP, and it being used as evidence for why they're good. Well, sure, a player who was luckier than most of his teammates in that when he happened to get his 130 hits for the season, 80 of those times a guy was on second and/or third, will be a 'good' player versus a player who only had a guy in scoring position 30 times. Thus, taking as evidence that BARISP is a good statistic the fact that 'good' players have a good BARISP is circular.
This is further discussed all over the internet; see this article by David Pinto, for another good take on the subject.