BABIP is a measure of balls actually hit that fall for a hit. if you have a high Linedrive % then a higher BABIP is expected and sustainable. if you don't then most likely you are blooping hits or getting grounders that are finding holes. That usually can't be sustained.
I haven't looked at Anthony's career numbers, but usually if a guy has a BABIP around .320 and they have a year that is .390, then you should expect regression. Only thing I would say is look at hard hit balls and line drive % to see if maybe there is a new approach or something clicked.
This is correct. BABIP helps determine BA, not the other way around. Your BA = BABIP + BABNIP (I made that acronym up, but it's HR and strikeouts mostly). BABIP is somewhat useful as a basic guide for all players (no one has ever sustained a .400 BABIP, for instance) but it is most useful when comparing a player to himself. Joey Votto has a career BABIP of .359; he has proven himself capable of sustaining a high BABIP. If his jumps to .400, it means he's probably been a little lucky (or he may have just been seeing/hitting the ball particularly well during that stretch, so you should expect it to drop but not terribly so. If he has a BABIP of .299, like he did two years ago, it means he's probably been quite unlucky and you should expect his numbers to improve.
Carlos Pena, on the other hand, had a career BABIP of .275. If he has a BABIP of .299, you should certainly not expect his numbers to improve, and should actually expect a bit of a regression. If he has a BABIP of .400, then he's almost certainly just been ridiculous unlucky and should plummet quickly. Chris Johnson has a career BABIP of .353. His BABIP of .400 in 2013 was not so out of the ordinary that it was unthinkable, but it was still an indicator of some luck, and everyone expected his .321 BA that year to drop again. And sure enough, his BABIP dropped to the more normal level of .345 the next year, and his BA dropped to .263. His BABIP was still high when comparing to the league as a whole that year, but it was not high for him, thus not an indicator of luck for him. For Pena, even that would have still led to an expectation of a further drop, but not for Johnson.
It is certainly possible for a player to suddenly become a better hitter and to raise his normal BABIP level. Again, there have definitely been players who have sustained a relatively high BABIP. If you tend to hit the ball harder than most and on a line more than most, you will also tend to have a higher BABIP. And even if a high BABIP stretch is temporary, that doesn't always mean it's driven by luck. It's possible for a player to suddenly see the ball better and hit it harder than he has typically done in the past. But that kind of stretch is still generally not sustainable for most players, so you should still expect it to drop if it's not in line with career norms.
Matt Kemp, FWIW, has a career BABIP of .340 and it sat at .297 this year. So it's definitely possible he sustains what he did with the Braves and has a year more in line with 2011-2014 next year.