Same word, same meaning, but "Kinder" (plural of Kind) gets an extra "n" added on in the dative case. Taken from wikipedia: "The dative plural of all nouns ends in -n if such an ending does not already exist, except that of nouns that form the plural with -s, which are usually loan words." http://en.wikipedia.org/wiki/German_nouns
Is it mean, that if it were Babys it should have been ""Ich spiele mit den Babys"?
Yes, "Babys" or "Babies" in dative. Either spelling is acceptable.
...does it mean.... ;-)
Edit: See wataya's post below
Just to clarify: the only correct German spelling today is "Babys". If you spell it "Babies" you're either using English spelling or one of the two pre-1996 spelling variants. That said, many people in Germany do use "Babies", but strictly speaking only "Babys" is correct.