Example:
We have a population of 100 people, where only 60 of them love pizza.
So, the probability of success is 0.6
SD[x] = $\sqrt{pq} = \sqrt{0.6 \times 0.4} = 0.48 $
The definition of SD is :
"Standard Deviation (SD) is the measure of spread of the numbers in a set of data from its mean value."
"What does the value of the standard deviation (0.48) tell us?"
What kind of spread are we talking about in this pizza example ?
If you take out one of the 100 people from your populations, then (s)he either loves pizza or (s)he does not, you do not know it in advance. If the person loves pizza then you assign a '1' else assign a zero.
If you do this many times then you will sometimes have a one and sometimes a zero, the standard deviation is the measure of variablity (spread) of these ones and zeroes.
Obviously, if 99% of the people in your population like pizza, then you expect a '1' most of the time, so the variability in the result will be lower ($\sigma=\sqrt{0.01 \times 0.99}=0.0995$).
This variability will be largest when $p=0.5$.
