Hot water heater thermostat contacts burned - how to repair?
So the first step is to find out why this terminal started burning in the first place.  Chances are you have a failed or failing element that is drawing too much power, but not quite enough to trip the breaker or you have a failed thermostat.

It looks like the main damage is the insulation of the wire that I assume goes to the supply.  You will need to replace or, if possible, repair the wire's insulation.  If the conductors are also burned you will need to replace.  Some heat-shrink tubing that has the same heat rating at the original wire, which is probably at least 90C could be used.  But check that as it should be marked on the undamaged wire.

A water heater element is a 240V connection that has no connection to ground.  Often these fail internally and one of the conductors will connect to the steel tube that contains the element itself.  So if you measure one with a VOM (that's a Volt-Ohm-Meter) with the POWER OFF!!! and the wires disconnected from the element, you should get a reasonable resistance (in OHMS mode) between the two terminals and a very high or infinite resistance from either terminal to ground.

What is reasonable?  Well you need to check the label on the unit where the element power outputs are listed.  Often they are in the range of 4-5 KW (that's kilo-Watts).  Ohm's Law and the Power Equation will give you the expected resistance:

P (in Watts) = V (in Volts) * I (in Amps)

So 4000 = 240 * I => I = 16.6 Amps, at 5000 W you have I = 20.8 A

Ohm's Law: V = IR which can be re-written as:

R = V/I

Again at 4000 we have R = 240/16.6 = ~14.5 Ohms and at 5000 we have 240/20.8 = 11.5 Ohms.  So you should see an R value on your VOM between the two terminals of the element in that range of 11.5 to 14.5 Ohms.  Much different than that (again assuming you have one that is 4000-5000 W) and you have a problem.