How does an MPPT charge controller control the input voltage?
An MPPT basically works by attempting to present a solar panel with the ideal "resistance".

In the following image, the red line shows the current-vs-voltage curve of a typical solar panel. 


The basic idea is that for zero current (open-circuit), they provide maximum output voltage, and as you increase the current output (i.e. you go from infinite resistance to zero resistance) the voltage of the solar panel decreases. Somewhere in this range, there is an ideal amount of "resistance" that when connected to the solar panel, will extract the most power. Or, in other words, there is an ideal amount of current to "pull" from the solar panel that will provide the maximum amount of power

The way MPPTs do this is by having some kind of DC-DC converter, such as a buck-boost converter, in which they carefully control the output current of the solar panel (i.e. the input current of the MPPT) such that the maximum power is obtained. Essentially, they behave as if they were variable resistors, trying to find the best spot to consume the most amount of power. Of course, this inevitable makes the output voltage variable, as this input current adjustment will come from the DC-DC converter's duty cycle.

Additionally, some MPPT algorithms are smarter than others, since sometimes there are multiple power maximums:


EDIT:

As requested, images were taken from here and here. Unfortunately these pages are not very informative.