He won't resign until it's clear there's an agreement among the other parties (which, in practice with these numbers, has to involve both the Conservatives and the Lib Dems) that would vote against him, and would be able to get their own programme, the "Queen's Speech" voted through in parliament.
But there are no formal rules about whether, or when, parties can negotiate with each other; so the Tories and Lib Dems can talk, and also say they will hold their talks first. And if that produces a definite agreement, then Brown will resign rather than get defeated the first time parliament reconvenes (which is Tuesday 18th May).
As the article says, Brown has said he recognises that the Lib Dems have the right to talk to the Tories first, since they got the most votes and seats, but has said he'll talk with the Lib Dems too, and has dangled a more concrete offer of an electoral reform referendum in front of them:
Clegg and Cameron are to meet tonight to discuss how their parties could work together in the future.
Brown recognized this in his statement today and said that Clegg and Cameron should "be entitled to take as much time as they feel necessary."
But Brown did not discount himself from participating in future government negotiations and went on to make his own overtures toward the Lib Dem leader, saying he was willing to discuss with him, "the areas where there may be some measure of agreement between our two parties."
One such area Brown highlighted is electoral reform, a key issue that was integral to Clegg's campaign, and something that Cameron in the past has said he doesn't want to change.
The thing is, the Tories and Lib Dems together can form a majority; Labour and the Lib Dems alone aren't enough, and they would need to have the Scottish and Welsh Nationalist parties in too. Which isn't too much of a stretch, in ideological terms, but short-term tactics and agenda can get in the way of complicated alliances, sometimes.