How many bits of information does a voltage of 0-5V , with accuracy of ±50mV convey?
What does accuracy of 50mV really mean?

It's an accuracy of +/- 50 mV and that means there is a spread of 100 mV. Put another way, you can't rely on a measurement being absolutely accurate so, if you measured a value of (say) 1 volt, the real voltage will be somewhere between 0.95 volts and 1.05 volts.

How did we come up with a division of every 100mV?

The +/- 50 mV inevitably sub-divides measurements into blocks having a range of 100 mV. That defines the error-free accuracy when converting the analogue voltage to a digital value.

What do 5.64 bits really show us?

It's a number that you can use to make comparisons. Many ADC specifications do this; an ADC may have 16 bit resolution but, at full scale and taking into account quantization and noise, the effective number of bits might only be 15.5. It's not a real number - it's a comparative guide number that tells you how good one ADC is versus another.