I know the following facts.  (Don't assume I know much more than the following facts.)


The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
The Atiyah-Singer index theorem can be proven using heat kernels.


This implies that both Riemann-Roch and Gauss-Bonnet can be proven using heat kernels.  Now, I don't think I have the background necessary to understand the details of the proofs, but I would really appreciate it if someone briefly outlined for me an extremely high-level summary of how the above two proofs might go.  Mostly what I'm looking for is physical intuition: when does one know that heat kernel methods are relevant to a mathematical problem?  Is the mathematical problem recast as a physical problem to do so, and how?  

(Also, does one get Riemann-Roch for Riemann surfaces only or can we also prove the version for more general algebraic curves?)

Edit:  Sorry, the original question was a little unclear.  While I appreciate the answers so far concerning how one gets from heat kernels to the index theorem to the two theorems I mentioned, I'm wondering what one can say about going from heat kernels directly to the two theorems I mentioned.  As Deane mentions in this comments, my hope is that this reduces the amount of formalism necessary to the point where the physical ideas are clear to someone without a lot of background.

Here is how the heat kernel proof of Atiyah-Singer goes at a high level. Let $(\partial_t - \Delta)u = 0$ and define the heat kernel (HK) or Green function via $\exp(-t\Delta):u(0,\cdot) \rightarrow u(t,\cdot)$. The HK derives from the solution of the heat equation on the circle: 

$u(t,\theta) = \sum_n a_n(t) \exp(in\theta) \implies a_n(t) = a_n(0)\cdot \exp(-tn^2)$

For a sufficiently nice case the solution of the heat equation is $u(t,\cdot) = \exp(-t\Delta) * u(0,\cdot)$.

The hard part is building the HK: we have to compute the eigenstuff of $\Delta$ (this is the Hodge theorem). But once we do that, a miracle occurs and we get the 


  Atiyah-Singer Theorem: The supertrace
  of the HK on forms is constant: viz. 
  
  $\mathrm{Tr}_s \exp(-t\Delta) = \sum_k (-1)^k \,\mathrm{Tr} \exp(-t\Delta^k) = \mathrm{const}.$


For $t$ large, this can be evaluated topologically; for small $t$, it can be evaluated analytically as an integral of a characteristic class.

Edit per Qiaochu's clarification

This article of Kotake (really in here as the books seem to be mixed up) proves Riemann-Roch directly using the heat kernel.
