DC current at input and ouput capacitance of buck
Don't confuse voltage with current, and charge/discharge currents (alternating directions) with a continuous current flow in one direction (DC). 
Your main problem here is misunderstanding of two similar words that have different meaning, namely IN and THROUGH.
You are right that DC current can't flow THROUGH the capacitor, but you are confused with the mentioned text about the current flowing IN(to) the capacitor.  
There is no contradiction here, as we all know that a current will flow INTO a capacitor when it is charging and will stop when it is charged up (meaning that the voltage across the capacitor is equal to the voltage of the source applied to the capacitor).
Likewise, the current will flow OUT of the capacitor when the capacitor's voltage is higher than the voltage of the circuit it is connected across, and will stop once they're equal.  





"That in turn means that only the ripple of the inductor current, centered around zero, flows through the capacitor. Then we have negative portions of the ripple that remove charge from the capacitor and positive portions of the ripple that add charge to the capacitor. Both must transfer the same amount of charge in equilibrium."
The "equilibrium" mentioned here is when there is an apparent constant voltage across the load and the capacitor in parallel and an apparent (somewhat constant) DC current flow through the load.  This apparent constant voltage is not perfectly flat, smooth, but it is a mostly DC with a small AC voltage on top of it (superimposed), so it keeps going up and down (its frequency is equal to the converter's switching frequency)."Centered around zero" - the "zero" here is the middle point of this small AC voltage on top of the DC voltage, in other words, the middle between its maximum and minimum value (which would be its positive and negative peaks if we isolate the AC component alone).The capacitor maintains the average, half-point or zero-point voltage between these maximum and minimum variations, and it does that by absorbing the charges/current (charging) during the maximum/peak point (voltage positive compared to the zero-point) and releasing the charges-current (discharging) during the minimum/valley point (voltage negative compared to the zero-point).The image below should illustrate my points here ("Mean, Vdc" is the same as the "centered around zero" or the "zero-point" mentioned above):



(The image is taken from https://learning.uonbi.ac.ke/courses/SPH307/scormPackages/path_2/44__waveform_filtering_smoothing1.html)