What does it mean to divide by the standard deviation?

I'm trying to variance normalise an image, in order to do so, I subtract the mean from the pixel value, to have a 0 mean, and divide by the standard deviation, to have a unit variance, right ? but I've also seen division by the standard deviation, since it's obviously not the same, what does it do ?

I'm also a bit confused about the values I get after normalisation, some values still more than 1.0 or less than -1.0, I thought a unit variance means a variance from -1.0 to 1.0 ?

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If you divide all the values by the standard deviation then you will then have a distribution with a standard deviation equal to $1$ (and so a variance equal to $1^2 = 1$). The difference is that the mean is not $0$, unless it was originally.
Except in a special case, the range will be more than twice the standard deviation: the special case is where the distribution only takes two values, and takes those with equal probability. So if you have a standardised distribution with mean $0$ and standard deviation $1$ then you will usually find some values outside the interval $[-1,1]$: for a standard normal distribution there is a probability over 31% that a value will be outside that intrerval.