statistical mechanics - Why is the canonical partition function the Laplace transform of the microcanonical partition function?

This web page says that the microcanonical partition function

$$\Omega(E) = \int \delta(H(x)-E) \,\mathrm{d}x$$ and the canonical partition function $$Z(\beta) = \int e^{-\beta H(x)}\,\mathrm{d}x$$ are related by the fact that $Z$ is the Laplace transform of $\Omega$. I can see mathematically that this is true, but why are they related this way? Why can we intrepret the integrand in $Z$ as a probability, and what allows us to identify $\beta = \frac{1}{kT}$?