It may be easier to consider a change in orbit caused by a pair of separate short thruster burns, to split the process into its various parts.
Consider a satellite in a perfectly circular orbit around the Earth. Its speed is constant, say $v_1$.
To go to a lower orbit, the satellite fires a thruster that opposes its motion. The satellite slows down, to a smaller velocity, ($v_2$) and is no longer in its original circular orbit.
Instead, it is now at the high point of a new elliptical orbit. As it continues in its orbit, it comes closer to the earth and speeds up. If it were left alone for a complete orbit, it would reach its maximum speed , $v_3$, a half orbit later at perigee, then climb back up until it reaches the high point of its new elliptical orbit again, with the same $v_2$.
Instead, the satellite fires its thruster again at perigee, opposing its motion and slowing down again from $v_3$ to $v_4$. With proper planning, this last velocity change leaves the satellite with the correct $v_4$ for a new lower circular orbit; this is a larger velocity than $v_1$!
The point is that the velocity increase during the coast phase is larger that the two combined velocity decreases in the two burn phases.
The same thing happens in reverse. Transferring a satellite from low earth orbit to geosynchronous orbit requires two burns. Both speed the satellite up, and the velocity in geosynch is lower than in LEO.
In the case of air friction, you have infinitely many "burns" slowing the satellite. It makes the math a bit more complex, but the prinicple is the same...