Why do we have to discharge the capacitor before testing it in an LCR Meter?
Both of the meters in question use some means of charging the capacitor under test by driving it with a controlled current and measuring the time it take to go between two voltages. When you charge a capacitor with a constant current the voltage rises linearly. (From \$ V = \frac {Q}{C} \$ we can differentiate to get \$ \frac {dV}{dt} = \frac {1}{C} \frac {dQ}{dt} = \frac{1}{C} I \$ since current, I, is charge per second. These circuits are typically powered from a 9 V battery and the actual test voltages will only be a couple of volts.

Capacitors, on the other hand, may be rated at much higher voltages - hundreds of volts - and if a charged capacitor, even if only a few volts higher than the test range, may destroy the test circuit. In practice a small charge of one or two volts shouldn't cause a problem other than a reading error.