Context: a person is trying to convince another that having other people worry about you is a good thing.
"Worrying" is like "letting other people behave freely". When you have people that worry about you, you can obediently behave like a spoiled child.
Of course my translation doesn't make much sense. I found on dictionaries that 甘える has two meaning:
behave like a spoilt child and
take advantage of. I don't think it is the second one, but the first one doesn't make sense either, how can you behave like a spoilt child in an obedient way? About 素直に, all its meaning are similar to
obediently. Can it be interpreted in a different way that I'm not getting here? I think that the general meaning of the sentence is "When you have people that worry about you, you can relax and do what you want (without worrying yourself)", but I'd like to understand the specific meaning of the words used. Please also correct the translation of the first sentence if I got it wrong. Thank you for your help!