From pure biophysical viewpoint the question
"Why does water buffer sudden temperature changes?" can be answered in the following way:
Water has relatively high specific heat capacity. This is the measure of the energy required to raise the temperature of one kilogram of a substance by one kelvin without a change of state occurring. "Relatively high" means that water can absorb more energy without noticeable (for the cell) increase in temperature.
Here are some heat capacity values from the above-linked Wikipedia article:
Animal (incl. human) tissue 3500 Jkg-1
Water at 25 °C liquid 4181.3 Jkg-1
Methanol liquid 2597 Jkg-1
Ethanol liquid 2440 Jkg-1
Paraffin wax solid 2500 Jkg-1
Graphite solid 710 Jkg-1
Water has relatively high thermal conductivity. This means that the absorbed heat is quickly distributed over the complete cell volume, leveling down the focal temperature increases in cell. Here are again some values for comparison:
Water 0.6 Wm-1K-1
Wood 0.2 Wm-1K-1
Paper 0.05 Wm-1K-1
Glycerol 0.3 Wm-1K-1
The follow-up questions:
"Why does water have relatively high heat capacity?" and "Why does water have relatively high thermal conductances?" are not within the scope of this site. However, here exactly comes your explanation with hydrogen bonds in play (from Wikipedia):
Hydrogen-containing polar molecules like ethanol, ammonia, and water
have powerful, intermolecular hydrogen bonds when in their liquid
phase. These bonds provide another place where heat may be stored as
potential energy of vibration, even at comparatively low temperatures.
Hydrogen bonds account for the fact that liquid water stores nearly
the theoretical limit of 3 R per mole of atoms, even at relatively low
temperatures (i.e. near the freezing point of water).