It's easy. A capacitor doesn't store charge, it stores energy. The net charge in a complete capacitor (rather than considering a single plate or the insulator) never changes. An increase of negative charge on one plate is exactly balanced by a decrease in negative charge on the other plate. Therefore, as current enters one terminal an equal current must leave the other terminal.
Your title asks about the value of the capacitors, and I think that has been adequately covered- you should match the nominal value to the specified load capacitance of the crystal (when in series with each other, and subtracting some allowance for input and stray capacitance).
The Q of a typical crystal resonator circuit is very high (maybe 100,000), and a small change in load capacitance won't affect the oscillation frequency by much. The equivalent "motional" capacitance of the resonator is quite high, so the pull effect of the load is small (typically measured in ppm/pF). If you are not using the crystal for a time keeping clock, it probably won't make much difference for you- it will vary with the crystal and load capacitance, but, say 5pF might make 30ppm or 100ppm difference in the oscillator frequency.
Since the capacitor might be 22pF, 5pF is a lot of change, so the tolerance and temperature coefficient is not very important. It's also cheap and easy to find almost perfect capacitors in the capacitance range used for load capacitors- ceramic NP0 types with tolerances of 5% are the cheapest and most available, and they're always rated for at least the voltage required (Vdd + 1.2V is certainly enough). Take the 27pF value- a Samsung CL10C270JB8NCNC is 5% tolerance, 50V, maximum drift of +/-30ppm/°C** and insulation resistance in the 10G range. All for $7.54 for a reel of 4,000 pieces, Digikey price. The difference between microwave and ordinary NP0 caps would not be noticed at 16MHz (except, of course, for the much higher price of the former). There are all kinds of complications (voltage coefficient, high temperature coefficient, microphonics, aging) associated with high value ceramic capacitors that don't apply much, if at all, to NP0 parts.
TL;DR Bottom line- if you use the most common NP0 ceramic capacitors in your favorite size, your circuit performance will not be limited by the capacitors in virtually all cases.
** Note that a 30ppm/K change of the load capacitor would likely contribute less than 0.1ppm/K change to the oscillation frequency (the temperature changes will be dominated by the crystal itself).
Best Answer
Capacitor is a charge reservoir. Switched-mode power supplies need to charge it first. Too large capacitors might make the internal power supply loop go unstable, which would create large voltage deviations across the capacitor and potentially burn it due to too large capacitor heating caused by its non-zero parasitic resistance called "ESR".
Capacitor do burn quite often. Actually, the aluminum capacitor failure is the most common failure mechanism in large motor drives! Motor drives and other power electronics (solar inverter, wind inverter, car battery charger, ...) exhibit very large current ripples at various frequencies. These ripple currents cause capacitor heating (ESR), which degrades the capacitor capacitance and further increases ESR. It's like a positive feedback. Aluminum caps have limited lifetime measured in thousands of hours. Their lifetime also decreases with elevated temperatures.
The typical way to mitigate this issue is to use multiple parallel caps (splitting the ripple currents) or using higher quality capacitors. These methods, however, tend to increase cost of the final product. The electronics industry is very cutthroat nowadays, which gives rise to designing for full functionality and reasonable failure rate but not a bit more.
Other answers also list good examples of how not only the capacitor can burn but how the large capacitor can cause other components to burn.