Ah yes, its a bromeliad pup! When to remove the pup is up to you. The longer you let it grow in its current state the faster it will reach maturity. So if your intention is to grow this pup into an adult bromeliad then I would say leave it be until it is about a fourth or third the size of the host plant.
Clipping the plant is rather simple. Remove some dirt around that portion of the plant. You can see there is a larger leaf that seems to be coddling the pup, it belongs to the host. Your going to want to peel that back far enough to reach the base of the pup, try not to rip it off the host. Then simply reach in with your scissors and cut the base of the pup away from the host. It should be rather obvious what portion of the plant belongs to the host and what belongs to the pup once you remove some dirt and peel that leaf back.
Now transplant the pup into its own pot and care for it in the same way as you did for the host! Be sure to fill in the dirt that you removed while making the cut. Don't be supersized when the host grows a new pup after you remove the old one.