Keep it dry. Not being made out of impenetrable plastic, bamboo toothbrushes require just a tiny bit more care than a regular toothbrush. Dry it quickly on a towel after each use, and that I store it upright in a clean glass container. This ensures the bamboo stays dry throughout the day, keeping the toothbrush in better condition longer.
Upcycle it. If you can reuse your worn-out toothbrush before you compost it, so much the better. Then compost it.
Better for you, better for the environment.