On a motorcycle or quad bike what makes the chain snap apart and fly off??

99% of the time, it is caused by poor maintenance, due to ignorance...

broken master link

Not using chain lube or having it too slack.

lack of lubrication...wrong chain/sprocket...to tight/loose

an old chain will do it as will poor adjustment. or a poor quaility chain. always buy the best you can and keep on top of maintenance. it will save you money in the end (and a leg)

It is worn out, or has not been properly maintained.

Keep it clean and oiled, and you should get several years out of a chain. If you run it dirty, or don't oil it, it may only last days.

poor maintenance. learn how to adjust and care for your chain before it flies off and wraps around your leg or you wreck.

Debris in the chain ?

I assume you check your chain + bike every time you set off ?

Be safe !!

It could be misaligned, make sure that the cogs are in line. It could be too tight or too loose, it may need lubricating, it could have had a stick or something caught in it. It could be a crap chain.

THE WEAKEST LINK OBVIOUSLY.

if you don't keep the chain lubricated then friction will cause the chain links to get hot and wear and snap, also if the chain is old it may have stretched and might need replacing along with the sprockets too

Have a chain too lose will cause it to flop around and snap as will having a chain too tight because metal/steel stretches and becomes weaker with undo wear and stress. And of course, lack of a quality lubricant will cause your chain to wear and snap. And the master link isn't necessarily the first link to break.

most quads have drive shafts never seen one with a chain they four wheel drive///? you got a rare one hang on to it.

if your chain is coming off there is either to much tension on it or it is too loose..

poor maintenance or using it beyond its normal life.

I see most common problem is the chain was too little play. When you jump or big pot hole, The swing arm move ( swing arm go to main frame on rear bike, Large travel) that make the chain get more tighter, and make snap. Even a dirt bike will break chain due to worn out chain link pins. or a rock ( good size) that caught between chain and sprocket. Will snap. other is very cheap chains for high performance bike, It will snaps. also I have seen when chain snap or fell off, The big problem is the chain will hit crankcase and break it. Buy better or best chain. save the bike from major damages. also If the master link clip in wrong direction. It will come out and the master link plate will come out then the master link will bend and fly the chain.

check alignment of the wheels