In the picture you supplied, there is a thing that looks like a C-clamp on the left. The typical way of hooking up a filter like that is to attach the clamp to the water pipe under your sink and turn the handle. This pierces the pipe and lets water flow to the filter.
The only problem with this arrangement is that when you move you will have to leave the clamp/valve behind and get a new one. Plus your lease may prohibit actually making changes to your plumbing.
You might be able to make an adaptor to connect the filter to the faucet, but everytime you turn the faucet off you will be running the risk of getting air into the system. I don't know if that is a problem with RO systems, but it's something to think about.
HTH