It basically boils down to the fact that machmeters hadn't been invented yet, because the physics of supersonic flight wasn't sufficiently understood back then.

Shock waves develop inside the pitot tube under certain conditions, and that causes a spike (or sometimes a drop) in pressure which is interpreted by the instrument as a corresponding rise or drop in airspeed.

Edit: as for the 262 claim itself it was pretty handily debunked in a post-war NACA technical note. If you dig around on the NTRS you can find it.

From Wikipedia: " Compressibility in pitot tubes of the time often resulted in exaggerated speed readings near the speed of sound, particularly in German equipment, which was adversely affected by supply shortages as the war progressed"

https://en.wikipedia.org/wiki/Hans_Guido_Mutke

In other words, it's the same problem that the piston-engined aircraft of the time had: Anomalous readings at high speed.