To simplify the explanation and terminology, let's consider the case of a spacecraft orbiting Earth.
All orbits are elliptical, with the center of mass of the system (Earth, in our example) at one focus. Circular orbits are a special case where the ellipse has no eccentricity and the focii are coincident.
The orbit of our spacecraft has a perigee (closest point to Earth) and apogee (farthest point from Earth). As an aside: the terms "perigee" and "apogee" are used specifically for orbits around Earth; around an unspecified object, we use the terms "periapsis" and "apoapsis". As our spacecraft completes one orbit around Earth, it will pass through perigee once, and apogee once.
Kepler worked out that an object in orbit (our spacecraft) will not travel at a uniform speed, but instead carve equal areas in equal time as it orbits. This means that it will have its highest speed at perigee and lowest speed at apogee. A truly circular orbit is a special case where its speed actually is uniform.
Let's consider a spaceship in a circular orbit. It wants to climb to a higher orbit. It briefly fires its engine to speed up along its current trajectory. It is now no longer in circular orbit; it is now at the perigee of an elliptical orbit, and as it travels to apogee, it must slow down. If it does nothing else, it will return to that same perigee point after one complete orbit. On the other hand, when it gets to the apogee point, it can fire its engine again to speed up again. This will raise the perigee point; the right change in speed at apogee will raise the perigee to make the orbit circular again, at the new altitude.
The "velocity increases then decreases" would refer to either an interpretation of Kepler's second law, or what typically happens when a spacecraft makes an adjustment to its orbit, such as I've described.