Do I have to put a shelf on top of it for maximum strength or will is still be just as supportive without a shelf?

Placing a shelf on it will actually lower it's strength by adding more weight to it that it must support. However, it should still reasonably hold weight on it. The real difference in how well it supports a closet rod (or a shelf for that matter) is how you mount it; if you just screw it into drywall it's really not going to hold well at all. However, screw it properly into a stud, using the right type of screws of the right length for the size of your studs and it should be pretty darn strong due to the support leg forming a triangle. I have no idea what the strength rating of this particular bracket is - it's really more of a consumer cosmetic bracket than one created for more serious use, but for household things on a small shelf like a few books, shoes, whatever then I would think that two of these properly mounted would be more than strong enough.