There are myriad benefits to a speedy trial, but 3 probably stand above the rest:
1. Preparation - the nature of the criminal justice system is such that those who are accused of crime usually are in the best position to prepare a case for trial. If a person takes appropriate steps up front, it is possible to get a leg up on the prosecution before they know what hit them. For example, in a typical jurisdiction, a prosecutor may be handling 10, 20, or even 50 cases at a given time. She can only devote x/20, or x/50 hours to your case. An attorney that you hire, and work with directly at the outset can focus on your legal matter, and the key issues in the case. By forcing a speedy trial, it is possible to gain a tactical advantage.
2. Memory - A speedy trial is important because, over time, memories fade about what happened at a certain location, or on a certain date. Law enforcement has the benefit of being required, in most cases, to generate reports about what actions they took. Over time, as your memory is waning about what happened, the cops remain sharp by taking a look at what they wrote down.
3. Custody - The right to a speedy trial is probably most important to those criminal defendants who are in jail. The constitutional, statutory, and common-law rights to a speedy trial help to guarantee, though they cannot always, that a person who has done nothing wrong doesn't spend too much time in jail before it is determined that they were innocent after all.
Every legal situation is different, and this answer should not be construed to apply to all cases. Contacting a lawyer about your particular situation is always best. A decision to rely on this answer in making decisions about a case establishes no attorney - client relationship.