I'm working in probability and statistics, and in that context I feel I have a reasonable understanding of the two-sided Laplace transform, $$ f^\star(\lambda) = \mathscr{L}[f](\lambda) = \int_{-\infty}^\infty e^{-\lambda x}f(x)\,\mathrm{d}x. $$ However, I don't have much knowledge of complex analysis, and so the general formula for its inverse, the Bromwich integral, feels rather opaque: $$ f(x) = \mathscr{L^{-1}}[f^\star](x) = \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} e^{\lambda x}f^\star(\lambda)\,\mathrm{d}\lambda. $$ Presumably, if I really want to understand the inverse Laplace transform, there's no substitute for studying complex analysis up to the point of being able to make sense of and evaluate such an integral.
However, as a tourist in the area, I'm wondering if there is some kind of intuition, geometrical or otherwise, that can help me understand why the second equation above would be the inverse of the first one. It would be helpful also to have some intuition about how the value of $\gamma$ should be chosen, and when and why the inverse exists and is unique.
I understand Fourier transforms and I have a reasonable intuition for complex numbers in general, but I haven't studied complex analysis and don't know how to integrate complex functions.