Using the definition of Laplace transform, we write
\begin{equation*}
e^{-as}F(s) = e^{-as}\int_0^{\infty}f(\tau)e^{-s\tau}\,d\tau = \int_0^{\infty}f(\tau)e^{-s(a+\tau)}\,d\tau\text{.}
\end{equation*}
Using the change of variable \(t=a+\tau\) and \(dt=d\tau\text{,}\) we obtain
\begin{equation*}
e^{-as}F(s) = \int_a^{\infty}f(t-a)e^{-st}\,dt\text{.}
\end{equation*}
Since \(U_a(t) \,f(t-a) =0\) for \(t\lt a\text{,}\) and \(U_a(t) \,f(t-a) =f(t-a)\) for \(t>a\text{,}\) we can write the last equation as
\begin{equation*}
e^{-as}F(s) = \int_0^{\infty}U_a(t)f(t-a)e^{-st}\,dt = \mathcal{L}\big(U_a(t)f(t-a)\big).
\end{equation*}