OK, the basics as I understand it is this...
High octane (98-101) fuel actually burns cooler and longer in the engine compared to standard fuel (94-95). This leads to cooler temperatures within the combustion chamber which prevent unburnt fuel left behind or in other cylinders awaiting ignition to be ignited out of sequence, which allows the engine to fire in the correct manor which maintains power. Also, as a smaller amount burns for longer thus releasing the same amount of energy, less fuel is needed per explosion, or another way to think about it (especially in performance engines) is that the same amount of fuel will produce more energy as it is combusted. Hence why high octane is considered a "performance" fuel.
I've seen a few car programs test out the difference in the different fuels, and they mostly concluded that in everyday, small engined vehicles, you should use the cheapest supermarket fuels as there's no difference in them. However, at the power end of the market (engines like the 2.0T-FSI engine) there is a noticeable difference of 5-10 BHP (at most) in engine output. Figures quoted are from memory and at the time of writing, I can not confirm they are accurate.
From my personal experience, I've been running Tesco Momentum 99-octane fuel now for a few thousand miles, along with my stage one tune, and have found that when driving around town and on motorways, I'm seeing more miles out of a tank of fuel. It's so easy to drive sensibly as the power is available lower down the rev range and allows you to take a more gentle approach to town driving. Torque is available from as little as 2,000RPM and means you can drive well below the turbo.
However, hit a quiet B-road or bit of motorway and boy do you appreciate having a re-map. Not only is the 0-62 time slightly improved, but it's the 30-80 in third performance that never fails to put a smile on my face. Although I've moved my front brake upgrade up the priority list, as it's a little scary when a corner appears faster than you're used too.
Enough rambling. Hope this helped!