How did they get a 650 horsepower engine into Ken Block's 2013 Ford Fiesta ST Gymkhana car?

People have been doing this kind of thing almost since the birth of cars. Forex, Jay Leno owns a mid engine, rwd V8 powered Ford Festiva, nee Shogun, that puts out a mere 300 horsepower. Jay Leno's Fantastic Ford Festiva: IIRC, it isn't a supertuned engine by any means, but it isn't on a custom frame as your example is, so pumping it up to 1000 HP wouldn't be practical unless ripping the rear of the car off the front is the desired goal.

Engine power output is dependent on airflow in and out of the engine. typically to get more power, it's easier just to increase the size of the engine by adding cylinders and bore size. however, you can keep a small engine by turbocharging it and working the head to get as much air in and out as possible by custom machine work. Of course, they're using top quality forged components in their build but it's not witchcraft. Lots of boost from the turbocharger combined with lots of fuel = lots of HP!

His car uses a 2.0 l Ecoboost. The engine parts are forged and machined to withstand heavy stress at high rpms. The engine uses forced induction by two heavy turbochargers which run at a high boost pressure. With all this tech the engine is able to produce insane amounts of power, which has to be restricted to 65o bhp so that the car can be more tamable and predictable on track.

The engine size is technically the physical size, they just increase the output of the turbocharging system and additionally tune the engine in various ways. You could technically tune just about any engine to output astronomical amounts of power, but with additional power comes additional stresses on the engine and the rest of the car. So, the longevity of the engine and the car itself will generally decrease as power is increased.

Ken block's fiesta uses a custom built engine that produces 838bhp, detuned to 641bhp. Because there is such a thing as too much power :p

Short answer, lots of things. Start off with pulling the engine. Rebuild all the internals with forged, high strength.components. Replace the Cam with a more aggressive timing and tuned to the power band you want.  Upgrade the injectors, spark plugs, fuel pump, intake, exhaust. Increase the turbo size, add a bigger intercooler. Polish and port anything that flows air, add a free flowing intake system. Then put a restricted on it because 830+ horsepower was too much to make it handle well.

650 hp out of a large 4 cylinder (I assume the engine in the Gymkhana car is at least 2.3L, much larger than the 1.5L in the production Fiesta) is very possible with a big turbo and tuned to run race gas.