How did Voyager 2 visit four planets on a single trip?

It used each planet as a slingshot to the next. A gravity assist makes use of the relative movement and gravity of a planet to change a spacecraft's path and speed, typically to save propellant. The planet does the work: any kinetic energy the spacecraft gains is lost by the planet, in line with Newton's third law.

That matters because the liftoff mass a mission needs grows exponentially with the delta-v it requires, so a probe that can borrow speed from a planet saves a great deal of propellant.

What made Voyager special was timing. In 1964 Gary Flandro, an engineer at the Jet Propulsion Laboratory, noticed a rare alignment of the outer planets that comes once every 175 years. It let a craft reach all of them using gravitational assists, and his Grand Tour idea cut the trip from forty years to less than ten. NASA cancelled the original, bigger plan in December 1971 when funding moved to the Space Shuttle, then proposed a scaled-down version in 1972: two spacecraft and four planets.

The two probes launched in 1977. Voyager 2 went first, on a path that allowed flybys of Jupiter, Saturn, Uranus and Neptune, and it visited them in 1979, 1981, 1986 and 1989. It remains the only spacecraft to have visited either ice giant. A similar alignment will not occur again until the middle of the 22nd century.