If we gloss over technical details Wiener process can be described as a sum of random draws from normal distribution - that is starting at 0 we keep adding a new random number at each time step. However we are going simplify this even further, and will use even more basic process - starting at 0, at each time step we flip a coin, and with heads we add 1, with tails we subtract one. And although this may sound like a grade-school version of Wiener process this is actually a valid "substitute" for our purposes.
Here I plotted some sample paths. And yes, this binomial walk looks quite like gaussian random walk.
Now we will look at the process in greater detail, and specifically we will look at the volatility of the process at each time step.
After 2 steps we have 3 values, and 4 distinct paths: +1 + 1 = 2, +1 -1 = 0, -1 + 1 = 0, and -1 -1 = -2. We use the same method to calculate variance: (2)2 * ¼ + (0)2 * ½ + (-2)2 * ¼ = 1 + 1 = 2. So, t=2, variance=2, volatility=√2 . Ok, let's do one more step:
After 3 steps we have 4 values, and 8 distinct paths which I am not going to enumerate; count for yourself. We use the same method to calculate variance: (3)2 * ⅛ + (1)2 * ⅜ + (-1)2 * ⅜ + (-3)2 * ⅛ = 9/8 + 3/8 + 3/8 + 9/8 = 24/8 = 3. So, t=3, variance=3, volatility=√3
As you can see variance scales with time, while volatility scales with its square root. I should point out that volatility is not the only quantity to scale with the √t, the property also holds true for expected high, expected low, expected range, expected drawdown and drawup - they all scale the same way, as I wrote in another post.
Finally, Vance writes that the relationship approximately holds for options (assuming no rates and no dividends). I would clarify: it holds approximately assuming small rates, small dividends, and small volatility. Also it holds much better for ATM options, but not so much when you move away from the money.