Truncated Gaussian distribution interpretation What is the interpretation/logic of this formula:

It's probability density function $f$, for $a<x<b$, is given by
$$ f(x;\mu,\sigma^2,a,b) = P(x|a<x<b) = \frac{p(x,
 a<x<b)}{P(a<x<b)}=\frac{p(x)}{P(a<x<b)} \\
 = \frac{\tfrac{1}{\sqrt{2\pi}\sigma} \exp\{-\tfrac{(x-\mu)^2}{2\sigma^2}\}}{\Phi(\tfrac{b-\mu}{\sigma})
 -\Phi(\tfrac{a-\mu}{\sigma})} = \frac{\tfrac{1}{\sigma} \phi(\tfrac{x-\mu}{\sigma})}{\Phi(\tfrac{b-\mu}{\sigma})
 -\Phi(\tfrac{a-\mu}{\sigma})} $$

EDIT: I was confused with what goes in the numerator and denominator and the rationale behind it in order to be able to explaint it to myself. I was reading about conditional probability P(B|A) formula and I was confused because I thought that numerator equals Probability of X being in the range [a, b] times probability of X given its in the range. 
Since I am learning this all by myself there are lots of things I am confused with and have no one to ask, except you guys here. As a result, this was the cause of a vague question at first.
 A: From the axioms of probability, total probability is $ P(\Omega) = P(-\infty < X < \infty) = 1 $. Let's denote probability of $X$ being in some subset $P(a < X \le b)$ as $\pi$. If $a > -\infty$ and $b < \infty$, then obviously $\pi < 1$. If your variable is truncated, i.e. it has restricted range, then it's total probability is some $\pi$, so we have to normalize it that it's total probability is equal to $1$, that is why we divide density function, or probability function, by $\pi$. It is a property of any truncated distribution, not just truncated normal.

I was confused because I thought that numerator equals Probability of
  X being in the range [a, b] times probability of X given its in the
  range.

You are correct! We are talking here about conditional probability of $X = x$ given that distribution of $X$ is truncated to the $(a,b]$ range since conditioning is about restricting sample space. That is what normalization of the probability by dividing by $\pi$ is about. So your formula can be re-writed as:
$$ \underbrace{f(x ~ | ~ a < X \leq b)}_\text{truncated density} = \frac{\overbrace{f(x)}^\text{non-truncated density}}{\underbrace{F(b)-F(a)}_{P(a < X \leq b)}} ~~ \text{for } a < x \leq b $$
