The inner tube won't 'explode' as long as it's entirely contained within the tire. But if part of the tire bead creeps out of the rim then the tube can expand through that opening and have a very sudden (and loud) rupture outside the tire. Likely causes are 1) improper mounting of the tire so that part of the tube gets pinched between the tire bead the rim, 2) uneven mounting of the tire with part of the bead lower in the rim channel and part of it higher and not fully seated, 3) a defect in the tire (least likely of the three). Tires should be able to handle pressures with a considerable safety margin over their rated pressure - 120 psi in a tire rated for 116 psi should not be a problem.

Try remounting the tire with a new tube. Make sure that the tube is entirely within the tire and no part is stuck between the rim edge and the tire bead. Inflate part way and check that the tire bead is at a uniform depth in the rim all the way around. Finish inflating to the desired pressure and again check that the tire is evenly seated all around the rim.