My question is related to the question Explanation for the Chern Character to this question about Todd classes, and to this question about the Atiyah-Singer index theorem.

I'm trying to learn the Atiyah-Singer index theorem from standard and less-standard sources, and what I really want now is some soft, heuristic, not-necessarily-rigourous intuitive explanation of why it should be true. I am really just looking for a mental picture, analogous somehow to the mental picture I have of Gauss-Bonnet: "increasing Gaussian curvature tears holes in a surface".

The Atiyah-Singer theorem reads $$\mathrm{Ind}(D)=\int_{T^\ast M}\mathrm{ch}([\sigma_m(D)])\smile \mathrm{Td}(T^\ast M \otimes \mathbb{C})$$
What I want to understand is what the Chern character cup Todd class is actually measuring (heuristically- it doesn't have to be precisely true), and why, integrated over the cotangent bundle, this should give rise to the index of a Fredholm operator. I'm not so much interested in exact formulae at this point as in gleaning some sort of intuition for what is going on "under the hood". The Chern character is beautifully interpreted in this answer by Tyler Lawson, which, however, doesn't tell me what it means to cup it with the Todd class (I can guess that it's some sort of exponent of the logarithm of a formal group law, but this might be rubbish, and it's still not clear what that should be supposed to be measuring). Peter Teichner gives another, to my mind perhaps even more compelling answer, relating the Chern character with looping-delooping (going up and down the n-category ladder? ), but again, I'm missing a picture of what role the Todd class plays in this picture, and why it should have anything to do with the genus of an elliptic operator. I'm also missing a "big picture" explanation of Fei Han's work, even after having read his thesis (can someone familiar with this paper summarize the conceptual idea without the technical details?). Similarly, Jose Figueroa-O'Farrill's answer looks intriguing, but what I'm missing in that picture is intuitive understanding of why at zero temperature, the Witten index should have anything at all to do with Chern characters and Todd classes.

I know (at least in principle) that on both sides of the equation the manifold can be replaced with a point, where the index theorem holds true trivially; but that looks to me like an argument to convince somebody of the fact that it is true, and not an argument which gives any insight as to **why** it's true.

Let me add background about the Todd class, explained to me by Nigel Higson: "The Todd class is the correction factor that you need to make the Thom homomorphism commute with the Chern character." (I wish I could draw commutative diagrams on MathOverflow!) So for a vector bundle $V\longrightarrow E\longrightarrow X$, you have a Thom homomorphism in the top row $K(X)\rightarrow K_c(E)$, one in the bottom row $H^\ast_c(X;\mathbb{Q})\rightarrow H_c^\ast(E;\mathbb{Q})$, and Chern characters going from the top row to the bottom row. This diagram doesn't commute in general, but it commutes modulo the action of $\mathrm{TD}(E)$. I don't think I understand why any of this is relevant.

In summary, my question is

Do you have a soft not-necessarily-rigourous intuitive explanation of what each term in the Atiyah-Singer index theorem is trying to measure, and of why, in these terms, the Atiyah-Singer index theorem might be expected to hold true.