How does the moon accelerate as it orbits the Earth?
Its velocity changes, which is the physics definition of acceleration.
Acceleration, in a physics format, does not necessarily mean that something is moving faster and faster - it also relates to a change in direction.
With the moon's orbit (and in fact any orbit, including Earth's orbit of the Sun), there are two competing forces: the force of gravity and inertia. The combined motions create the motion of the blue line in the diagram - the orbital path (which, in essence, is the falling of the object in orbit but never landing).
With the orbit being a circle, the velocity of the object is continually changing, and so the Moon experiences continual acceleration.