Why can't it be "Les mères aiment ses enfants"?
because its multiple (les) mothers/subjects liking multiple (leurs) children/ objects.
If you used 'ses' it would be correspond to one subject liking multiple objects that belong to her........i.e the mother likes likes her children.
From quick googling, it seems that "leurs" is for when the subject (i.e. the mothers) is plural, while "ses" is for single subjects (e.g. a mother).
Perhaps someone else can correct me or give a better explanation.
That's exactly the reason. "ses" would be used only if you had a singular subject.
Without the context, I get the impression that "Des meres comme leurs enfants" (I'm not sure if this is correct) referring to mothers being like their children instead of the mothers liking their children should be accepted.
Why is "meres aiment leurs enfants" wrong?
Because in French you never use a noun without an article before (there might be exceptions, but I can't think of any at the moment).
Well, yeah, apart from things like beaucoup de and things like that. If you're saying "things" in general you use les.