Best Answer - Chosen by Voting
1. It means:
"I have many hobbies".
持っています=>Have
If you break the word and just think of the first Kanji "持", it means, to 'hold'....etc. So, "持っています" means that you have something that BELONGS to you.
2. It can be used just like English sentences...For example, when someone asks, "How many hobbies do you have?" //OR// "Do you have any hobbies?", you can reply saying->はい、たくさんのしゅみを持っています！ (I think this reply fits well with the second question!)
Hope my answer was helpful ^_^
