Because we need to use the possessive form to translate correctly.
Therefore, it's "son eau".
Well on the normal audio it totally sounds like "le". I can't say for the slow audio as I don't have access to it.
The difference of pronunciation between "le" and "leur(s)" is the "r" sound, and also the "e" sound is a bit more open for "leur(s)". Note that the "s" at the end, if there is any, is not pronounced.
Try this to get used to the difference (don't mind the translation, it's just for the audio) :
I don't understand why it could be translated as "The dog is drinking HER water." The word water is feminine anyways, isn't it?
"son" can mean "his" "her" or "its" depending on the gender of the person who owns the water (which in this sentence could be anyone, or anything, even the dog istelf).