You're probably familiar with the interpretation of the $z$ score as "the number of standard deviations the test statistic is from the mean". This interpretation can be obtained from the formula for the $z$ score.

Consider,

$$z = \frac{x-\mu}{\sigma},$$

if we multiply $\sigma$ to the left hand side we get,

$$ z \sigma = x-\mu,$$

here we see that $x-\mu$ is equal to the standard deviation $z$ times.

There is a similar interpretation for your variable $x'$.

The definition of $x'$ is,

$$ x' = \frac{x-\mu}{\text{range}},$$

we can once again multiply the range to the left hand side obtaining,

$$ x' \text{range} = x-\mu.$$

Here we see that $x-\mu$ is equal to the range $x'$ times. In other words $x'$ is the fraction of the range which makes up the distance between $x$ and $\mu$.