Why can the effective number of bits be a non-integer? What does this physically represent?
It is a result of calculating the number of bits after the imperfections such as noise and distortion are taken into account.
An ENOB of 6.8 basically tells you that for example an 8-bit ADC has real world performance that is better than an ideal 6-bit ADC but worse than an ideal 7-bit ADC.
You can also think of it as not having 2^6=64 or 2^7=128 discrete steps in the signal like on ideal 6 or 7 bit ADC, but 2^6.8 or about 111 discrete steps.