/u/gilareefer explained how the mechanism works, but to answer your question more generally, this type of switch mechanism is used because it's very resistant to oxidation/corrosion, and because a traditional mechanism used with a bimetallic strip would be prone to arcing.

The temperature-sensitive part is a bimetallic strip shaped in such a way so as to "tilt" one way or the other based on temperature due to the difference in the metals' thermal expansion coefficients. A simple switch mechanism based on this might be to just allow the strip (or a contact attached to the strip) to touch another contact in a certain position; but this type of switch is prone to corrosion over time due to being exposed to the elements. The complicated liquid-metal-drop-inside-sealed-vial mechanism prevents corrosion because there's no oxygen inside the vial.

Additionally, a simple switch contact mechanism used here would likely cause arcing and damage to the contacts since the switch would open and close extremely slowly. Things like normal light switches prevent this by integrating a "snap" mechanism to very quickly open and close the circuit; but a bimetallic strip supplies very little force, not enough to operate such a snap mechanism. The liquid-metal-inside-sealed-vial mechanism solves this problem by a) surface tension keeps the drop of mercury together so once the vial tilts even a little the whole drop will quickly slide over and complete the connection; and b) any damage due to arcing would be minimized in a low-oxygen environment inside the vial.

For both of these reasons, mercury metal is used because it's the only metal that's liquid across the usual range of human environmental temperatures, so is the only metal suitable for use in this type of switch mechanism.