On television it was not explained how the Master survived. However, there have been various accounts given by other media to explain this survival.
In "The Glorious Dead" (Doctor Who Magazine comic story), it is explained that he was throw around inside the Eye of Harmony/Vortex, until he ended up at the place where the various multiverses converge. He was able to escape from here, and possessed the body of a homeless preacher.
In the Eighth Doctor Adventure novels it is revealed late in the series' run that the Master has been inside the TARDIS all along, and that the Doctor sometimes speaks to him. (This was originally intended to lead to the Android Master in "Scream of the Shalka" (webcast).) Presumably, he is resurrected from here.
(It is extremely difficult, and perhaps impossible, to reconcile all three of these accounts.)