Why does power help us maintain altitude in slow flight?
It is not correct to say that power controls altitude. In slow flight, it is taught that power controls altitude and pitch controls speed. Yes, it works, but you have to always keep in mind that a power change will always require a pitch change and a pitch change will require a power change, if you want to maintain the same trimmed speed, altitude and angle of attack.
As you might very well know, in a slow flight, you reduce your power first and start putting down the flaps to sort of keep the aircraft in an approach configuration. Then you pull back on the stick until you are close to the stall speed. Once the speed you want is reached, you add power as required to maintain speed and altitude. If you lose altitude, a little addition of power will arrest it. But trust me, this will also increase the speed by small margin. This might not be registered in the air speed indicator of a smaller aircraft. Similarly, an increase in pitch in attempt to reduce an increasing speed, will change the altitude. So, it is not fixed. Pitch and power are not two separate entities. They always work together. Try flying a high performance aircraft with powerful engines and with a load of inertia and you will see how true my words are.
The reason why more and more power is necessary to successfully accomplish a slow flight is because you are operating below the L/D speed of the aircraft. Once you are below this speed, the increase in induced drag (due to high angle of attack) increases the total drag on the air frame. To overcome this drag you are required to add power or energy. We call this operating behind the power curve or the region of reverse command. Reverse because normally, you need an increase in power to fly at a higher speed. But behind the curve, to fly at a lower speed, you need more and more power.
Look at the graph below, the GD (Green dot) speed is the L/D speed. As you can see, as you go below the L/D there is a steep increase in drag.