"My mother is going out to eat."
Translation:मेरी माँ खाने बाहर जा रही हैं।
There is a subtle difference between the two. "मेरी माँ खाने बाहर जा रही हैं" means "My mother is going out to eat" while "मेरी माँ बाहर खाने जा रही हैं" "My mother is going to eat outside".
However, in spoken Hindi, you can convey either meaning with both sentences depending on which words you choose to stress.
निकलना is more like 'to leave'.
मेरी माँ खाने निकल रही हैं। - 'My mother is leaving to go eat' is stressing the action of leaving the house. It doesn't make a whole lot of sense in the present continuous tense but other tenses like 'मेरी माँ अभी-अभी खाने निकली' (My mother just left to go eat') are more natural.