Why is the canonical partition function an exponential? It makes intuitive sense that micro-states of higher energy occur with a lower probability and the exponential function has reasonable properties. However can a physical explanation be given to why the energy's are distributed according to an exponential function? 
 A: It is a probability distribution, so physics (equations of motion...) alone is not enough. Probability of some state is determined with help of probability theory, which is useful in physics.
The Boltzmann distribution is the only function that is consistent with general assumptions about systems analyzed in statistical physics (in the meaning probabilistic methods in physics, the ordinary meaning of word statistics does not apply much there) with definite temperature $T$.
The usual assumptions about large systems with definite temperature are:
Probability part:


*

*probability of state being in some very small region of state space is proportional to product of size(volume) of that region and probability density in the region (the probability density changes little for little changes of state). 

*Sum of all probabilities is 1.
Physics part:


*energy of system is not constant, but is exchanged between the system and the reservoir (at the same temperature $T$) irregularly and for the supersystem containing both is conserved all along;

*subsystems can be assumed to be independent in the sense that probability that (subsystem A has state 1 and subsystem B has state 2) is product of probabilities (probability that A is in 1) and (probability that B is in 2).
The only class of functions that obeys the assumptions 1, 3, 4 is $Ce^{-\frac{E(state)}{k_B T}}$, where $C$ is real constant. Its value is fixed by 2.
The function does not apply well to all physical systems, because there are systems that do not obey above assumptions, for example galaxy or globular star cluster is a mechanical system, but its interacting subsystems do not behave independently and cannot be reasonably assigned the same probability function as the big system.
For more details on the derivation for ordinary systems where the assumptions apply well, see Landau Lifshitz, Statistical physics I. For the examples of systems where it does not work so well, search statistics of gravitational systems, or keyword gravothermal catastrophe.
