How does the ski shape of an aircraft carrier deck help for take off?
The Ski jump was originally used by the UK Royal Navy to assist Harriers in getting airborne.

The Harrier cannot take-off vertically when fully loaded.  In all models, the thrust is lower than the maximum weight.  Therefore, they need to do a "short take-off" where the thrust nozzles are angled rearwards to provide thrust, then partially angled downwards as speed is gained to produce enough lift from the wings, and thrust angled downwards, to climb away.  Watch the nozzles closely at about 17 and 36 seconds in this video.  Try not to watch the poor chaps forehead too closely at 20 seconds.  Also watch this video.  At about 16 seconds, you can see the pilots left hand pull back on the nozzle angle lever as he passes the end of the deck.

Now imagine a flat deck on an aircraft carrier.  Operations must be able to carry on in rough seas.  You accelerate down the deck and fly off the end, just as the deck dips downwards as the carrier rides a large wave.  You are pointed straight at the sea.  In order to avoid flying into the sea, you need to climb, which means pulling G and a lot of power at low speed which the Harrier just didn't have.  

Now consider the ski jump.  There are two effects.  


The jump is angled so that even in the worst seas in which operations will continue, the bow pitching down will not point an aircaft leaving the end of the jump towards the sea.  The aircraft will always be pitched upwards.
Imagine driving a fast car up the jump and off the end.  It will be thrown up and forward in a parabola.  Even a car, which cannot fly, will be pointing up for a few seconds, then level for few seconds before falling into the sea.  But instead of a car, you are driving a jet which can fly and has thrust. 


The aircraft will angle the nozzles rearwards for maximum acceleration until it leaves the jump. The wings are now generating significant lift, let's say 50% of that required to climb away.  As you leave the end of the jump, you angle the nozzles downwards so that you have the momentum of the aircraft and some rearward thrust maintaining lift on the wings whilst adding lift from the thrust vector having a downward component.  

Those seconds of pointing up, then level, enable the aircraft to continue to accelerate until all of the lift required can be gained from the wing when the nozzles can now be angled rearward again to continue the acceleration. Take-off speed is therefore reduced and take-off distance is about halved meaning smaller, cheaper carriers.

That's how Harriers do it.  I imagine it's the same story for any other VSTOL aircraft.