Denote by $A$ the connection and by $F_A$ its curvature. Then
$$dA=F_A-A\wedge A. $$
If $A$ is in Coulomb gauge we have an additional equation
$$d^*A=0. $$
The advantage is that the operator $d\oplus d^*$ is elliptic and now we have an equation of the form
$$ (d\oplus d^*)A= \mbox{something}. $$
Elliptic theory allows us to convert bounds on "something'' into bounds on $A$. Then, the bounds on $A$ can be converted into compactness results using standard compactness results in Sobolev spaces.
Finding a local Coulomb gauge on a region $D$ is possible as long as the "energy" $\Vert F_A\Vert_{L^2(D)}$ is smaller than $<\epsilon$, where $\epsilon$ is related to the second Chern number of a principal $G$-bundle over $S^4$, the conformal compactification of $\mathbb{R}^4$. The energy of an instanton on $S^4$ is equal, up to a universal constant, to the second Chern number which is an integer. You can regard this as a quantization result, stating that the energy of an instanton is an integral multiple of a universal constant. (If my memory serves me right this constant is $4\pi^2$, give or take a factor of $2$.)