SiC Avalanche Voltage
The avalanche voltage is the actual voltage where a diode starts to break down while reverse-biased. Using a typical SiC Schottky diode with a voltage rating of 1200V, the avalanche voltage would be above the voltage rating (e.g. maybe 1400V at room temperature). If you increased the voltage across the diode to the avalanche voltage, the diode would start conducting large amounts of current (breaking down). 



(From wikipedia: https://upload.wikimedia.org/wikipedia/commons/thumb/0/07/V-a_characteristic_Zener_diode.svg/500px-V-a_characteristic_Zener_diode.svg.png)

The avalanche voltage is often not specified in the datasheet, as the manufacturer "just" needs to make sure that the avalanche voltage is always greater than the rated voltage. In this case, a particular diode may have an avalanche voltage of 1400V at 25°C, but maybe only 1300V at the maximum rated temperature of 175°C. Unless you have built a circuit specifically to operate at/near the avalanche voltage, exceeding the avalanche voltage usually results in massive overcurrent events (high voltage + high current) and exploding devices.

The leakage current is the (relatively) small current that flows when the diode is reverse-biased and the voltage is below the avalanche voltage. Picking a random SiC diode, the leakage current when the diode is reverse-biased at 1200V is typically 30µA at 25°C, but it increases to a typical value of 55µA at 175°C. 

I'll note that the trends that they mention are also usually true for most semiconductors.