When a nucleus decays why do the particles fly apart at an extremely high speed? And what holds the particles together in the nucleus of a stable atom?

The strong force holds them there. This is what gluons "communicate". So they are going to move fast because like a strong spring that breaks, the strong force holds them tightly; the neutrons and protons are very close and the protons are repelling each other, hence when they are able to break free of this, or there is decay, this gives rise to fast expulsion speeds