reconstruction filter - How does it actually work?
The sampling theorem requires a perfectly bandlimited signal, bandlimited to below twice the sampling frequency.  The problem with this is that only an infinite length signal (e.g. exists before the big bang) can be perfectly bandlimited.  This is from the Fourier theorem regarding any domain with finite support.  

Thus all real-world signals are imperfectly band-limited, limited by the length of the recording session and the duration of the impulse response of the low-pass filtering used before sampling (e.g. imperfect band-limiting).  So you must assume a finite noise floor due to sample aliasing.  Therefore your reconstruction also need be no better than this noise floor.  Therefore you can window the perfect reconstruction formula to some reasonable finite duration.  For audio, this window width can be limited to on the order of a small multiple of the period of the lowest frequency audible, as stuff before and after that can't affect audible pitch perception significantly (other stuff, such as masking and adaptive thresholding, etc., likely dominate).

The advantage of very high sample rates is that the aliased frequencies (due to imperfect pre-sampling filtering and post-sampling reconstruction) are even higher, and far less likely to be picked up by a physical microphone or pass with any significant magnitude (above the thermal noise floor, etc.) through the anti-aliasing filters.  As other answers here detail, this also allows physically realizable filters to be flatter and have more linear phase response in generic 20-20k pass-band, and roll-off more approaching half the sample rate.