The experiment you just performed, originally conducted by Frederick Griffith in 1928, demonstrated a phenomenon called bacterial transformation. This is the process where bacteria can alter their genetic makeup by taking in DNA from their surroundings.
The key finding was that the mice injected with a mixture of harmless live RII bacteria and harmless heat-killed SIII bacteria still died. When a sample from these mice was examined, living SIII bacteria were found.
This result showed that a "transforming principle" from the dead SIII cells had been transferred to the live RII cells, giving them the ability to produce the protective capsule and become deadly. We now know that this transforming principle is DNA, the molecule that carries genetic information.