The Robotics Today talk by Scott K is a fantastic overview of what they're doing at BD.

These sorts of approaches are nothing new; they've been published on by other legged robotics groups for 5-10 years and are based in optimal control theory which is now many decades old. The reason they've become more popular is because these methods are computationally-intensive, so until recently it wasn't feasible to compute whole-body motion control over a time horizon in real-time (meaning at 100-1000Hz).

BD's secret sauce is likely 1) really really incredible hardware - it cannot be overstated how important this is, as most legged robots don't even have the power to execute a jump, let alone not break themselves on landing) 2) really good models and 3) very smart usage of model-based control methods which take advantage of said really good models.

I guarantee that many academic labs across the world have the theory in place to do similar things as BD, but BD's hardware is years ahead of anything non-commercial and they have a massive team of people working on putting out these demos. Their work is incredible, but no one should underestimate the insane amount of work that goes into their videos.