Wet a wick, whirl it through the air, and the water evaporates. Evaporation costs latent heat, and the only place to take it from is the bulb underneath — so the thermometer falls, and keeps falling until the heat arriving from the air exactly balances the heat leaving as vapour. Where it settles is the wet-bulb temperature $t_{wb}$.

It is the lowest temperature evaporation alone can reach, which is why it — and not the dry bulb — is the number a cooling tower, an evaporative cooler and a heat-stress limit are all written against. A tower's cold water can APPROACH the wet bulb and can never reach it.

Read the instrument with the psychrometric equation: $\varphi = \dfrac{p_{ws}(t_{wb}) - A\,p\,(t_{db} - t_{wb})}{p_{ws}(t_{db})}$. $t_{db}$ is the dry-bulb reading and $t_{wb}$ the wet-bulb reading, both in °C; $p$ is the barometric pressure in pascals; $p_{ws}(\cdot)$ means the saturation curve evaluated at whichever temperature is in the bracket — it is called twice, at two different temperatures, and mixing them up is the classic slip. $A$ is the psychrometer constant, $6.66\times10^{-4}\ \mathrm{K^{-1}}$, and it is an INSTRUMENT property, not a constant of nature: that value is the WMO's for a properly ventilated psychrometer with air moving at 3 m/s or more over the wick. A stagnant wall-mounted hygrometer runs nearer $8\times10^{-4}$ and reads humid. That is why you whirl it.

Going the other way — wet bulb from dry bulb and relative humidity — has no closed form in the physics, so Roland Stull fitted one in 2011 and it reproduces a full psychrometric solve to about 0.3 K. Use it between −20 and +50 °C and between 5 and 99 % humidity, and keep the ordering in view as your sanity check: dew point ≤ wet bulb ≤ dry bulb. A wet bulb that lands below the dew point is arithmetic, not weather.