To convolve two functions we have the following:
$ y(t) = h(t) * x(t) = \int_{-\infty}^\infty x(t)h(t-\tau)d\tau $
Plugging in functions for x(t) and h(t) we get:
$ = \int_{-\infty}^\infty e^{-\tau}u(\tau)u(t-1-\tau)d\tau $
We now change the interval of integration to reflect the step function $ u(\tau) $
$ = \int_{0}^\infty e^{-\tau}u(t-1-\tau)d\tau $
Finally, by changing the interval of integration once again we get:
$ \begin{align} &= \int_{0}^{t-1} e^{-\tau}d\tau \\ &= -e^{-(t-1)} - (-e^{0}) \\ &= 1 - e^{-(t-1)} \end{align} $
