What you have pictured is not a freewheel. It's a cassette. The difference is that a freewheel has a ratcheting mechanism in it that lets you coast, and it screws onto a hub; a cassette slides onto a hub that has that ratcheting mechanism built into it, and is retained by a lockring. So you can't substitute a freewheel for a cassette or vice-versa; you want to make sure you get the right thing.
You need to make sure that the rear derailleur will accommodate the bigger sprocket, and that it has enough capacity to wrap up the additional amount of chain (23 teeth vs 15 teeth). Capacity probably won't be a problem; maximum sprocket size might be, although it if's close, you can fake it. You should be able to look up the derailleur on the manufacturer's website to find this information. You'll need a replacement chain, as the chain needs to be longer to fit around the bigger sprocket.
To make the swap, you'll need a long-handled crescent wrench (you need to apply a lot of torque to remove the lockring and put it back) and a lockring removal tool, which looks like a big bolt with a splined tube on one side. You'll need a 7-speed chain (chains are speed-specific), and you'll probably need a chain tool to break the existing chain and to drive the rivet to close the new chain. Or you could take it to a shop.