Well, the first thing to note is that High Speed Trains are not actually that new – the Shinkansen has been around since the sixties, for example.
As far as the technologies that made fast trains possible in general, though, I would say it was mainly down to improvements in track and catenary technology and the invention of the yaw damper.
In large part, the innovation with the fastest trains is not with the trains themselves, but with the supporting infrastructure. Building track with very wide corners and smooth gradients, ‘superelevation’ (leaning the corners inwards like a motorcycle), and using rails that are continuously welded together in sections more than a thousand feet long rather than much shorter sections joined with fishplates for a smoother ride, to give a few examples. It was also important to develop high tension catenary (overhead wires) and special pantographs that could handle rubbing against those wires at very high speed while conducting power to the locomotives.
It was also important to develop electronics capable of handling high voltage and high current within the confines of a locomotive body. Pretty much all post-steam locomotives drive their wheels with electricity, but for diesel (or older gas turbine, etc) locomotives there’s an on board generator hooked up to the engine. Moving all of that electrical generation outside of the locomotive and only carrying the motors (and some conversion electronics on board) allows more power in the same space and weight. That’s why the fastest diesel trains are only capable of ~145mph while the TGV’s wheeled train speed record is >350mph.
Another important development for the Shinkansen was something called a Yaw Damper. Essentially, train wheels are cone shaped – this is what keeps them on the rails, and allows them to go round corners even with a single fixed axle even though the outer wheel is covering a longer distance. The downside to this is that at high speed, trains have a tendency to rock from side. This is called Hunting Oscillation, and in the best case this can cause a lot of noise and friction as the wheel flanges impact the track while at worst it can cause serious damage to the track and wheels.
The solution to this was a combination of less conical wheels and the invention of the Yaw Damper, a kind of shock absorber used to reduce Hunting Oscillation.
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