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what is this phrase mean? たくさん　の　しゅみ　持っています

From my opinion: it means someone has some hobbes.

yes，it means somebody has a lots habbies.
たくさん = a lot

しゅみ = hobby

持っています = have,has

when someone asks [ご趣味は]＝'what's your hobby',you can say[たくさんのしゅみ持っています]

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 ^_^