According to the Wikipedia page on the Act, the Royal Navy declared slave traders pirates and set up a squadron in the Atlantic to seize them. By the way, in that case could British privateers also capture slave ships?
From my (very limited) understanding of prizetaking, a captured ship would be taken to a Prize Court in a British dominion, where they would condemn the vessel if it was a lawful capture and award the value of the prize as bounty, right?
What would happen to the (presumably now freed) slaves after that? Did they just stay where the Prize Court was located or were they transported somewhere?
They tended to be released in Sierra Leone. One of the prize courts was in Freetown, in Sierra Leone, and land in Sierra Leone was allotted to the freed slaves. The Americans had a similar system based in Liberia, where slaves found on condemned slavers were similarly allocated land. Jim Walvin's "Black Ivory" (and his many other books on the trans-Atlantic slave trade) is quite good on this aspect.