The first stage must provide a lot of thrust to get into lower gravity, so it's quite heavy with fuel and has 9 engines. After the fuel in that stage is expended, the rocket splits in two, dropping that first stage back to earth. The top half of the rocket contains additional fuel and whatever payload the rocket is delivering (including potentially an additional stage + engine) and is powered with a single engine.
The first stage that drops off is the one that SpaceX is attempting to recapture. Historically, that stage just lands in the ocean and is either scuttled or must be extensively cleaned and retrofit in order to fly again. SpaceX figures if you leave a bit of extra fuel in the tanks, you can land it vertically at the same site it took off from, and save yourself the time and expense associated with cleaning and retrieving it.
This would be very valuable since the rocket engines are typically the most expensive components of a launch, and the first stage has 9 such engines:
It isn't higher gravity they're overcoming with the first stage, but rather the much higher atmospheric pressure near the surface. Gravity in orbit is almost identical to gravity at the surface.
