"Why would you cool down to later compress (and heat up again)?"
Because the efficiency of any combustion engine depends on the temperature differential between the input and the output. It cannot produce more energy than that differential (second law of thermodynamics) so cooling the input allows the engine to produce more energy, assuming the output remains the same.
At Mach 6 it's just too hot and energetic to compress and stabley combust, as I understand. Recall that stagnation temperature is a function of the square of velocity. The reddit article says the compressor 'puts heat back in' but that's a byproduct rather than the aim. The aim of the compressor in this engine is to get enough moles of oxygen in the combustion chamber with enough moles of hydrogen to combust and produce enough thrust. You couldn't compress the air sufficiently to squeeze it into the combustion chamber if you didn't cool it a bit first.
This is my understanding based on reading what's publically available on the net anyway.
They had to develope a special fuel for the SR 71. One with very high flash point. I imagine that the temperatures at mach 6 are too high for normal fuels too.
Yep. Skunk works describes how they pumped it around the cockpit to help cool the aviation electronics and pilot. Just to add to the scariness of flying at mach 3+, you're also surrounded by jet fuel.
They had to develope a special fuel for the SR 71. One with very high flash point. I guess that the temperatures at mach 6 are too high for normal fuels too.
Sounds like a shotgun solution to the passer by such as myself
Edit: it is a shotgunsolution... They just added an old turbine after the cooler component to test it.