Why doesn't the Tsetse fly wipe out all animal life in its range?
Firstly, as Remi.b points out, although Trypanosoma brucei infection is invariably fatal in humans, the same is not true for all host species. Domestic animals, particularly cattle, are an important reservoir of T. brucei rhodesiense and can also be infected with T. brucei gambiense (although it's not clear how important a reservoir they are for gambiense).

Secondly, just because all hosts in an area get bitten by tsetse does not mean they all become infected with trypanosomes. Even in endemic areas most vectors never become infectious. The transmission cycle for trypanosomes and other insect-borne pathogens requires an incubation period in the insect vector which is often longer than the median lifespan of the insect, so even if all insects get infected early in life, most will never survive the incubation period to become infectious. The later the average time of first infection gets, the higher the fraction of hosts that will have the opportunity to reproduce before dying.

Finally, it isn't entirely accurate to say "there is no way to prevent being bitten"; it is possible to reduce exposure to insects behaviourally. For example, in most endemic areas local residents will know which areas are most heavily infested and should be avoided.