The force that causes similar charges to repel each other is called the electromagnetic force. Sometimes the energy of the nuclei is so high that they can get close enough for the nuclear strong force to be stronger than the force that repels similar charges. The strong force is what binds protons and neutrons together in the nucleus. It's very strong at small distances but almost non-existent when the distance between the particles is bigger than their size. For nuclear fusion to work, the nuclei must get close enough that the strong force is greater than the repelling electromagnetic force.
Higher temperature means higher velocities means harder collisions.
Fusion more likely to occur at high temperatures. The high pressure in the center contains the reaction, so that everything doesn't just escape. In the Sun and other low mass stars, nuclear fusion happens by the proton-proton chain. Hydrogen nucleus is just a lone proton, but helium is 2 protons and 2 neutrons. This missing mass is converted to energy and released during the reaction.