The Boost is not a second run. It gives the run you just started a bigger budget, so it pulls twice as many name ideas from the same brief. The offer shows up right after you buy Domain Finder. One click adds it to the same card, with nothing to re-enter. If you say no, or the charge doesn't go through, your run starts as normal.
Is it worth it? It comes down to how much you want to choose from. If your brief is narrow and you already know the direction, the standard run is often enough. If the field is wide open or you have a lot of constraints, more ideas give you a better shot at a name worth building on.
You can add the Boost until your run finishes, or until it has used most of its budget. If you still see the offer on your project, it's still open. Domain Finder and the Boost are both covered by the 14-day money-back guarantee: one click in Billing, the full amount back to your card, no questions.