Not a general election, no.
The Prime Minister is technically chosen by a vote of the House of Representatives (like the US we have a two chamber legislature and we 'borrowed' the American names for those chambers). In practice, this means the Prime Minister is virtually always the leader of the party that has the largest number of seats in the House of Representatives, of which he is one Member.
In this case, that's the Liberal Party - and it's held an internal vote to change its leader, and therefore that also changes the Prime Minister. It's perfectly constitutional and has happened quite a few times in our history.
Technically, Mr Abbott could refuse to resign until the House of Representatives had voted, but there's no point. He'd lose such a vote. So he will resign, and the Governor General will commission Malcolm Turnbull as Prime Minister.
Yes, exactly. Thank you.