They feel the current combination of tyranny and social chaos are responsible for much of the suffering in the world. What anarchists fear is the corrupting influence of power and the inevitable abuse of power. An individual can only do so much damage, but the same person in a position of authority, or worse yet, an organized, systematic application of corrupted power, can wreak horrible damage. And note that the police only prevent crimes in rare situations, such as when a police officer just happens to be at the scene of a crime in progress. The police almost always show up after the crime has been committed.
Most crimes go unsolved.
Attempting to punish offenders after they have committed their crimes is not a very effective way to protect people. The government loudly proclaims its mandate anyway. Most of the objections people have to anarchism as a social system are based on the assumption that people are unreasonable and irresponsible. If this were the case, no amount of police, judges and jails could conjure order out of chaos. Much of the destructive behavior we suffer from is committed by individuals who have been raised in the most dire conditions, and who face very limited personal choices due to the material and cultural poverty they were raised in.