Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins of this construction from gauge theory on surfaces, a la AtiyahFloer conjecture, which I have then forgotten. What is the origin of Heegard Floer?
3 Answers
I think the crude answer is that there is (or maybe just should be) an extended 4 dimensional TQFT that assigns the Fukaya category of a symmetric product to a surface, and the usual HeegardFloer Lagrangian to a 3 manifold. So, the usual definition of HeegardFloer is the gluing formula for a Heegard splitting, and invariance is no miracle at all.

1$\begingroup$ Yes, of course. Denis Auroux gave a talk about it wwwmath.mit.edu/~auroux/papers/slidesfuksymg.pdf and it does remove the miraculousness. But this was a posteriori, in light of LipshitzOzsvathThurston. Surely this is not how OzsváthSzabo came up with this? $\endgroup$– Max MNov 2, 2009 at 20:58

1$\begingroup$ I think it's a caricature of their thought processes. This TQFT is supposed to come from gauge theory which was known at the time. In particular, I believe that the symmetric power showed as a space of some solutions to equations. I would say the point of that work of Auroux is that you can explicitly reconstruct the higher levels of the TQFT from the Heegard Floer theory. $\endgroup$– Ben Webster ♦Nov 2, 2009 at 21:20

$\begingroup$ I think you are right. I was hoping to get some info on the gauge theory. From what I gather, the idea is that the symmetric product is the space of solutions of vortex equations  explained arxiv.org/abs/math/0606063 This is U(1) gauge theory,and presumably "monopole" version of AtiyahFloer is what produces the Heegard Floer, which OzsvathSzabo then went on to study directly. I wonder if anyone can flesh out some details (e.g. how to the Lagrangian tori arise). $\endgroup$– Max MNov 3, 2009 at 2:03
From Szabó's delightfully understated response (pdf) to receiving the Veblen prize:
The joint work with Peter Ozsváth which is noted here grew out of our attempts to understand Seiberg–Witten moduli spaces over threemanifolds where the metric degenerates along a surface. This led to the construction of Heegaard Floer homology that involved both topological tools, such as Heegaard diagrams, and tools from symplectic geometry, such as holomorphic disks with Lagrangian boundary constraints. The time spent on investigating Heegaard Floer homology and its relationship with problems in lowdimensional topology was rather interesting.
Of course, if one believes that Heegaard Floer homology is somehow the limit of monopole Floer homology as one degenerates the metric in some way that depends on the Heegaard diagram, then the independence of Heegaard Floer homology from the Heegaard diagram would fall out from the metricindependence of monopole Floer homology. Unfortunately, I can't seem to find references that give any sort of precise picture of how Ozsváth and Szabó came to think that this should be the case (though it might have been a baby analogue of the picture in this paper^{link broken} (pdf) by YiJen Lee, written a few years later).
It perhaps bears mentioning that Heegaard Floer homology wasn't the first invariant that Ozsváth and Szabó constructed based on thinking about the interaction of the SeibergWitten equations with a Heegaard diagram—see The Theta Divisor and ThreeManifold Invariants and The theta divisor and the Casson–Walker invariant, which extract an invariant from the thetadivisor of the Heegaard surface, appear to have been based on thinking about what happens to the Seiberg–Witten equations when one has a neck $S\times [T,T]$ ($S$ is the Heegaard surface) with the metric on $S$ at $t=T$ itself having long cylinders over the compressing circles for one handlebody, while the metric on $S$ at $t=T$ has long cylinders over the compressing circles for the other handlebody.

$\begingroup$ The link to the paper of YiJen Lee is broken, can you remember what it was meant to be? $\endgroup$– David Roberts ♦Nov 6, 2021 at 3:46

I'm far away from being an expert, but I think the Heegaard Floer homology was invented by Peter Ozsváth and Zoltán Szabó, so I would recommend the following link to you: click me
If this Introduction is not enough, you should perhaps read "the original work" (in fact the Heegard Floer homology was developed in a long series of papers): P. S. Ozsváth and Z. Szabó. Holomorphic disks and topological invariants for closed threemanifolds. To appear in Annals of Math., math.SG/0101206.
EDIT: Perhaps the Introduction of the book Floer homology, gauge theory, and lowdimensional topology is useful if you are interested in the motivation of Heegard Floer homology.

$\begingroup$ I think the link to the clay article might be this one: web.math.princeton.edu/~petero/Introduction.pdf $\endgroup$ Oct 12 at 17:23