Consider the following proposition
Proposition C
Let $Z = F(X)$; where $F$ is the continuous cumulative distribution function of the random variable X, then $Z$ has a uniform distribution on $[0, 1]$.
Proof
$P(Z \leq z) = P(F(X) \leq z) = P(X \leq F^{-1}(z)) = F(F^{-1}(z)) = z$
This is the uniform cdf.
I can follow the proof above, but my interpretation of its meaning isn't making sense to me. This proof seems to imply that the cdf of any random variable has a uniform distribution. Is this correct? Is there an intuitive explanation for why this is?
For example, consider the graph of the cdf of some normal distributions from wikipedia

How would you map the ideas of this proposition to the graph of the cdf above?