here I have to prove the fourier transform of $e^{-a|t|}$ , the beginning of the proof is to rewrite $e^{-a|t|}$ as: $e^{-at} U(t) + e^{at} U(-t)$, I know how to continue the proof starting from this step but I don't understand why does the two forms above are equal?
Here, the functions $U(t)$ and $U(-t)$ are the unit step functions.
When $t$ is positive, $U(t)$ is 1, and $U(-t)$ is zero, so you get only the first term, which equals $e^{-a|t|} = e^{-at}$. Now do the same thing when $t$ is negative... (I'm assuming here that your book/prof has the definition that $U(x) = 1$ for $x \ge 0$ and is zero otherwise.)
(You might reasonably be worrying about the case $t = 0$, but if your function and the rewritten function differ at only a single point, they're equal in $L^2$ and hence will have the same Fourier transform.)