Andreyevskie ponds were dug in the period from the end of XVIII century through the first quarter of XIX century, as it is mentioned on Vorobyovy Hills Park website, and this fact on their history is the only one that can be found in the whole internet, so we don't know the real purpose of their creation (I'll update the decription if I find any additional info; I would also appreciate greatly any visitors' contibution), but we may try to guess it.
By the end of XVIII century Andreyevsky monastery became a women's workhouse, so it is unlikely that it was able to complete such project (though the ponds have been surely named after the monastery). But approximately at the same time (in 1756 namely) Neskuchny Garden was founded as a suburban recreational area (the area was outside Moscow boundary that time), and it is rather likely that Andreyevskie Ponds appeared here due to a plan to spread the recreational zone farther.
Two centuries later Andreyevskie ponds still serve recreational purpose. In 2006 the surrounding area was reconstructed, fish and waterfowl were settled here (unfoturnately I'm not sure whether you'll meet any of these nowadays) and an ecological path through Vorobyovy Hills starts right near the point where this micro-cache is hidden.