Turtles, like other reptiles, can endure long fasts, in their ability

Here's the version created via the use of (A).

Turtles, like other reptiles, can endure long fasts, in their ability to survive on weekly or even monthly feedings; however, when food is readily available, they may eat frequently and grow very fat.

That version conveys the meaning that turtles can endure fasts in their ability to survive. There's no way to endure in an ability. So, the meaning conveyed by the (A) version is nonsensical.

Here's the version created via the use of (B).

Turtles, like other reptiles, can endure long fasts, having their ability to survive on weekly or even monthly feedings; however, when food is readily available, they may eat frequently and grow very fat.

GMAT Ninja explained what's wrong with (B) quite well a couple posts above. I can add another point.

It does really make sense to call the basic ability to survive on weekly or monthly feedings "their ability," not only because it is clearly an ability had by turtles, so "their" is unnecessary, but also because, as the sentence even implies, turtles are not the only creatures that have that ability. So, it is actually "an ability" that a number of creatures have, not "their ability," which only turtles have.

On another note, a funny thing about this sentence is that even the correct version conveys a silly meaning. The fact that turtles can endure long fasts is pretty much the same as he fact that they are able to survive on weekly or even monthly feedings. So, the sentence basically says, "turtles can endure long fasts because they can survive long fasts." LOL.