Because, simply stated, an aircraft will only be stable in a specific configuration at a single speed and/or center of gravity. Any change in speed and/or center of gravity will make the plane want to climb, descend, roll, yaw, etc. Trim allows the pilot to decrease his workload by making fine-tuned adjustments to control surfaces that will "balance" (for lack of a better term) the controls for the specific configuration he's in. Ultimately, a perfectly trimmed aircraft will fly "hands-off". For maneuvering flight, adjusting trim makes certain control inputs easier--i.e. they require less force from the pilot, reducing the effort required, and reducing fatigue over the long term.
Since speed is a function of pitch, the elevator trim is often the most used trim control in the aircraft, since the most common configuration change in an aircraft is speed. Even the most simple planes will typically have pitch trim. Some more sophisticated aircraft have pitch, aileron, and rudder trim. Some WWII aircraft had all three, and some did not. The Bf-109 series, for example, did not have adjustable rudder trim tabs. Aileron and rudder trim are typically only seen on higher performance aircraft.