Why does my varistor cause my fuse to blew up?
The MOV is probably toast. They fail shorted. You can measure it with an ohmmeter. 

The rating (normal operation) is 250VAC (assumed to be RMS voltage of a sine wave), which is \$\sqrt{2}\cdot 250 \$, or about 354V peak.  

The 390V refers to the (nominal) peak voltage when the MOV is just starting to turn on (1mA). It's not clamping much, but it could get hot if that condition persisted, especially with DC. It could be as much as 429V or as little as 351V. 

When it is clamping a heavy transient (50A) the voltage can be as much as 650V. 



So if you anticipate surges no greater than 50A you can use it to protect a circuit that is capable of withstanding 650V peak. 

Perhaps worth noting that MOVs "wear out" and a series of heavy transients can lead them to failing shorted. Of course if the surge is large enough they can fail open 
because the lead wires have been blown off etc. 

Silicon devices such as bipolar TVS devices can provide sharper clamping without the wear-out mechanism (they will still fail- often shorted- if overloaded, of course).