I agree with Chad and Peachy, the sentence is awkward. It should read, 'what does she object to most about supermarkets'.
I think your question is, why are there two prepositions, 'to' and 'about'?
In this case 'to' is not a preposition, it is part of the infinitive, 'object to'. An infinitive is a verb plus 'to'.
'About' is a tricky word with several meanings. It can mean 'approximate', as in; he was about forty miles from home, or the apple cost about a dollar. It can also mean 'referring', as in; we talked about the weather, or he wrote an essay about eagles.
Your own sentences could read, 'I found this sentence in a book', and, 'could somebody explain this sentence to me please'.