First, this is from the end of Plato's dialogue, Phaedo, the last of Plato's treatises that talk about the end of Socrates' life. Second, it also important to note that one of the main topics of discussion in Phaedo is whether or not the soul is immortal. Third, where's your translation from? "Owe" I think misconstrues part of the meaning, I've always seen it as "offer", which makes a lot more sense if you're trying to give something to a god.

These last words have been subject to debate for a while, but in general the suggestion may be that since Asclepius was the god of healing, it ought to suggest that it is thanks for healing Socrates' soul, which is important because Plato (by way of Socrates) argues that the soul is immortal. At the same time, it may be a little bit of a slight to the Pythagoreans at the time, such as Simmias and Cebes, both from Thebes, who argued that people shouldn't harm animals because the souls of our ancestors may be in them. Socrates' counter-claim to them would likely be that only the body of the animals is harmed, while the soul (since it is immortal) remains intact.

This is only the general argument, immediately obvious in the context of the rest of Phaedo. For a more in-depth study see, for example...

1. Emily Wilson, The Death of Socrates (Harvard, 2007)

2. Colin Wells, "The Mystery of Socrates' Last Words," Arion: A Journal of Humanities and the Classics (Fall 2008): 137-148

3. J. Crooks, "Socrates' Last Words: Another Look at an Ancient Riddle," The Classical Quarterly 48 #1 (1998): 117-125

4. Glenn W. Most, ''A Cock for Asclepius,'' The Classical Quarterly 43 #1 (1993): 96-111

Hope this helps a little. Happy reading!