dítě is an irregular word which is neuter in the singular but becomes feminine in the plural: své dítě / své děti
I just discovered this by reading the Wiktionary entry: https://en.wiktionary.org/wiki/d%C3%ADtě#Czech It was not obvious to me before.
To quickly reinforce the reason behind the correct answer for pronouns, it would be really nice if Duolingo provided hover help that showed each noun's gender... or even better, what you said: "you provided the neuter version, but dítě is feminine" as an error message. For me and I imagine many others, getting these all synced up correctly has easily been the hardest part of learning Czech so far.
Right now, fully understanding why the answer is wrong, or what the right answer is, takes a little too much time poking around through wiktionary.
Can it not mean we raise our children? We help them get older? Starame = something about old.
No. Staráme has nothing to do with starý, though sometimes many centuries ago there might have been some connection. Starat se is to take care of something. It does not have to be a person, you can take care of a building, flowers, animals and yes, people.
Is svoje used more frequently than své in real life like moje and tvoje are based on my experience?
What is real life? There are so many layers and means of communication. Speaking with family members and friends is only a small part of functions of a language although it is the one everyone sees first.
Check http://syd.korpus.cz/NKqY5g6f.syn left is contemporary written Czech, right is contemporary spoken Czech.