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Gauss's law states ∮E⋅dA = Q_enclosed/ε₀ for any closed surface. This elegantly encodes the inverse-square nature of Coulomb's law and shows how enclosed charge alone determines flux, independent of external charges.
From electric flux and the divergence theorem, you have two key tools: you know that flux through a surface measures how much field "passes through" it, and you know that the divergence theorem converts a surface integral into a volume integral over the divergence of the field. Gauss's law ties these to the physical source of electric fields — charge — in one compact statement: the total electric flux through any closed surface equals the total enclosed charge divided by ε₀.
The deep reason this works is the inverse-square law. Electric field from a point charge falls off as 1/r², while the surface area of a sphere grows as r². These two factors cancel exactly, so the flux through a sphere of radius r is the same as through a sphere of radius 2r or 10r — as long as the same charge is enclosed. Equivalently, if you deform the sphere into any other closed shape, the flux still doesn't change, because no field lines are "created" or "destroyed" in the empty space between the charge and the surface. External charges contribute equal and opposite flux in and out, canceling exactly. Only the enclosed charge has an unbalanced contribution.
The Gaussian surface — the closed surface you choose — is a mathematical tool, not a physical object. The genius of Gauss's law is that you get to choose the surface. For a point charge, a concentric sphere is the natural choice because E is constant in magnitude and perpendicular to the surface everywhere, so the integral ∮E⋅dA collapses to E × 4πr². Setting this equal to Q/ε₀ immediately reproduces Coulomb's law. For a uniformly charged infinite line or plane, you choose a cylinder or pillbox respectively — again exploiting symmetry so the integral becomes trivial.
This is the key lesson of the integral form: Gauss's law is always true, but it is only *calculationally useful* when the charge distribution has enough symmetry that you can find a surface where E is constant in magnitude and either perpendicular or parallel to dA everywhere. When such symmetry exists, finding the field is an algebraic step, not an integration. When symmetry is absent, Gauss's law still holds but doesn't simplify the calculation — you'd need to fall back on Coulomb's law or the differential form (∇⋅E = ρ/ε₀) and other tools.