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 $2$.)