To make it more intuitive lets look at what is going on more abstractly!
We have two transformations, one for inputs and one for problems.I will denote both of them by $pad$, it will be clear from the context when it is the first one and when it is the second one.
These two transformations have the following property:
I. for all problems $A\subseteq \Sigma^{ * } $, for all inputs $x\in\Sigma^{ * } $:
$pad(x) \in pad(A)$ iff $x \in A$,
II. if $A$ is in $EXP$ ($NEXP$), then $pad(A)$ is in $P$ ($NP$).
III. the transformation for inputs is in complexity class $EXP$,
It is clear that the transformations for padding have these properties.
Now, the reason that we don't know how to do the same thing in the reverse direction is that we don't have transformations like padding in the reverse direction (when we exchange $EXP$ with $P$ and $NEXP$ with $NP$). So the question is why?
I don't have a formal argument why there are not such transformations at the moment, but intuitively what András Salamon said is correct. It is easy to increase the size of inputs, but it is not clear how they can be compressed?
Another way to understand it is to think about in the following way. Assume that $P=NP$, and we want to solve an $NEXP=NTime(2^{n^{O(1)}})$ problem. We are given an input $x$ of length $n$, we think of it as an input of length $N=2^{n^{O(1)}}$:
$NEXP(n) = NTime(2^{n^{O(1)}}) = NTime(N) \subseteq NP(N) \subseteq P(N) = Time(N^{O(1)}) = Time(2^{n^{O(1)}}) = EXP(n)$