The integrand is dominated near the maximum x* of f(x). Expanding f to second order gives a Gaussian integral. The result is exact for Gaussians and asymptotically exact for all smooth functions. For complex contour integrals (saddle-point method), the contour is deformed through the saddle.
Stirling's approximation (n! ≈ √(2πn)(n/e)ⁿ) is exactly the Laplace method applied to n! = ∫₀^∞ xⁿe^(-x)dx. Used in statistical mechanics (partition functions), quantum field theory (path integrals), Bayesian model selection, and asymptotic combinatorics.