The crystal is operating in the series resonant mode, while the oscillator needs a parallel resonant feedback network (180 degree phase shift at resonance, gain peak at resonance; whereas series resonant to ground would be gain minima). The capacitors act to give a series-parallel transformation, thus giving the desired characteristic.
The network's characteristic impedance is approximately the reactance of the capacitors. This allows some tuning to be done, e.g. using a somewhat lower ESR crystal on a weak oscillator (perhaps STM family MCUs?). Mind that the capacitance also has a small effect on the tuned frequency (give or take a kHz or so), so this will also affect exact frequency. (If capacitance is being reduced, additional capacitance can be connected in parallel with the crystal to maintain the same total capacitance as seen by the crystal; or if increased, some can be connected in series with the crystal.)
You might think to use a crystal in the parallel resonant mode, shunting to ground, with a series impedance to supply it, thus making an impedance divider which has gain peak at resonance; but this has 0 phase shift at resonance (for a resistive supply, or 90 for an inductive one), so won't work with the simplest oscillator (an inverting amplifier).
Tim