Having a simple majority of seats in the house of commons means that you won. All the new MPs vote for the prime minister, who forms a government. With a majority of a party, the MPs from that party vote for their choice of prime minister, who is the leader of their party. I am about 99% certain that to qualify to be PM, you have be a member of the house of commons, too, but 1% of me says maybe they can also be a lord (more on this later)