Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came from.
We take an orthogonal frame bundle $P$ of $TX$, a $\textrm{spin}^\mathbb{C}$ structure $\tilde{P}$ with determinant line bundle $\mathcal{L}$, the complex $\pm$ spin bundles $S^\pm(\tilde{P})$ associated to $\tilde{P}$, a unitary connection $A$ on $\mathcal{L}$, and then BAM:
$F_A^+=\psi\otimes\psi^*-\frac{1}{2}|\psi|^2$
$D_A\psi=0$
for a spinor $\psi\in C^\infty(S^+(\tilde{P}))$. From here we can consider the space of solutions (monopoles) and do some Floer theory stuff and whatnot.
I only read that these equations come from Witten's famous paper Monopoles and 4-Manifolds (along with two others joint with Seiberg)… however, unless I am mistaken, he simply writes them down and starts arguing for their similarity/duality to Donaldson's theory (with instanton solutions). I then try and go to the standard references of Donaldson, which don't seem to suggest how the SW equations come about (nor do I even really see how the instantons come about). Although I have studied physics for a long time, I seem to just juggle around these papers, without ever finding a natural "blooming" of the SW equations.
Even if it's in the language of String Theory, I would like to know the general story / understanding of the "blooming" of the SW equations, and how exactly they are "dual" to the instanton-scenario of Donaldson, perhaps even for the "blooming" of these instantons. (For instance, I don't see a set of equations for instantons). This post may not be stated in its clearest form, but I will try my best to make appropriate edits.
Best Answer
After thinking, and reading other references and re-reading the papers I mentioned, I may have found a sufficient explanation (at least to my care): Both instantons/monopoles are solutions to corresponding equations of motions from associated actions, and they "bloom" from an overarching SUSY action.
Witten formulated "twisted N=2 Supersymmetric Yang-Mills", a TQFT with SUSY (supersymmetry), which leads to the Donaldson invariants. This used an $SU(2)$-bundle over $X$ along with a gauge field (connection $\omega$) and matter fields (bosonic $\phi,\lambda$ and fermionic $\eta,\psi,\zeta$), and gave the Donaldson-Witten action functional $S_{DW}=\int_Xtr(\mathcal{L})$,
$\mathcal{L}=\frac{1}{4}F_\omega\wedge(\ast F_\omega+F_\omega)-\frac{1}{2}\zeta\wedge[\zeta,\phi]+id^\omega\psi\wedge\zeta-2i[\psi,\ast\psi]\lambda+i\phi d^\omega{\ast d^\omega}\lambda-\psi\wedge\ast d^\omega\eta$.
This has associated partition function $Z_{DW}=\int e^{-S_{DW}/g^2}D\Phi$ (here $\Phi$ denotes the space of aforementioned fields), where $g$ is a coupling constant that is the key here for answering our question. The "blooming" of this action functional is beyond the scope of my intentions and probably of MathOverflow, so I won't question it.
In weak coupling ($g\rightarrow 0$, known to physicists as the ultraviolet region), the action localizes to the classical Yang-Mills $S_{YM}=\int_Xtr(F_\omega\wedge\ast F_\omega)$ and have the equations of motion $d^\omega\ast F_\omega=0$. The global-minima solutions are $F_\omega=\pm\ast F_\omega$ (as Oliver clarifies in a comment). These solutions are the Donaldson instantons.
Now apparently, when we instead look at strong-coupling ($g\rightarrow\infty$, known to physicists as the infrared-region), the Seiberg-Witten equations should arise (a "duality" in Witten's TQFT). Indeed, Seiberg and Witten showed that this infrared limit of the above theory is equivalent to a weakly-coupled $U(1)$-gauge theory (the $SU(2)$-gauge group is spontaneously broken down to the maximal torus). Perhaps here is where a better understanding would be desirable (buzzwords 'asymptotic freedom' and 'symmetry breaking' appear).
Anyway, some physics-technique stuff happens (the previous paragraph can be described as "condensation of monopoles"), and we must consider a spin-c structure (which all of our oriented 4-manifolds have, whereas a spin structure would not allow us to consider all 4-manifolds); note that $Spin^c(4)=(SU(2)\times SU(2))\times_{\mathbb{Z}_2} U(1)$. This gives the data: $U(1)$-gauge field $A$ and positive spinor field $\psi$ (as written in the original post). The pair $(A,\psi)$ is a monopole when it minimizes an action $S_{SW}$, i.e. are time-independent solutions to equations of motions (the Seiberg-Witten equations). The action here is $S_{SW}=\int_X(|d^A\psi|^2+|F_A^+|^2+\frac{R}{4}|\psi|^2+\frac{1}{8}|\psi|^4)$, with scalar curvature $R$.
I hope this post is not too confusing.
[[Edit/Update]]: I just came across a book chapter by Siye Wu, The Geometry and Physics of the Seiberg-Witten Equations. These lectures tell the physical origin completely! (i.e. completely details my sketch). The SW equations and action functional pops up on pg191. http://www.springerlink.com/content/q37322037j466218/