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.
It must be discharged because it might be charged to high enough voltage to break the meter or high enough voltage to be hazardous to humans. LCR meters do not use high voltages, it might only use up to 1V signals for testing.
Thinking about the dangers of a charged capacitor, I remembered my school years when "bad guys" charged capacitors to the network and then gave them to the unsuspecting girls to hold them a little in their hands:) Others hid a small battery, an inductor and a switch in a "book" with metallic covers ... and again handed it to someone asking him/her to open it... to read something interesting inside:) I tell these "funny" stories to my students to arouse their interest in the behavior of these exotic electrical elements...
Both they accumulate electrical energy and can be thought as of "rechargeable batteries": the capacitor as a kind of "rechargeable voltage source" and the inductor as a kind of "rechargeable current source". It is interesting to see what of them is more dangerous (for people and devices).
The capacitor is not so dangerous if there are no high voltage sources around since it stores the same voltage. But it keeps the voltage for a long time - enough to carry it elsewhere and discharge it there. So a capacitor that is charged to high voltage is dangerous for a relatively long time.
In contrast, an inductor is dangerous even when it is charged with low voltage. But it is dangerous only in the first moment when we disconnect it from the source. That is why there is no warning for discharging an inductor... because when we connect it to the meter, it will already be discharged...