I guess that this is as explicit and low-tech as it gets: if $X$ is a K3 surface (i.e. a non-singular quartic hypersurface in $\mathbb{CP}^3$, with the complex orientation), then $X \# \overline{\mathbb{CP}}{}^2$ and $3\mathbb{CP}^2 \# 20\overline{\mathbb{CP}}{}^2$ are an exotic pair.
To see that they are homeomorphic, we use that odd indefinite forms are diagonalisable and then Freedman's theorem. (Ok, and that complex projective hypersurfaces are simply-connected, by Lefschetz's theorem.) To see that they are not diffeomorphic, we use that Kähler surfaces have (some) non-zero Seiberg–Witten invariant, while anything written as a connected sum of indefinite pieces doesn't. (In particular, the same argument applies to any hypersurface of $\mathbb{CP}^3$ of degree at least 4; in this case, blowing up/connected summing with $\overline{\mathbb{CP}}{}^2$ is only needed in even degrees.)
This is just the tip of the iceberg of the tip of the iceberg that Anubhav mentioned in his comment.
EDIT (06/03/2024): Levine, Lidman, and Piccirillo's paper [LLP] fits the bill. They give explicit constructions of exotic 4-manifolds using Kirby diagrams (and explicitly computing some Heegaard Floer diffeomorphism invariants).
While I'm at it with the edits... About the example I mentioned above: one can give explicit handle diagrams for the K3 and for its exotic companion (see Gompf and Stipsicz's book), so those examples where explicit too. There's an important difference with [LLP]: the original computation of the SW invariants of the K3 and of $3\mathbb{CP}^2 \# 20\overline{\mathbb{CP}}{}^2$ was done using properties of SW (an something similar with HF), while in [LLP] they compute the invariants from the handle decomposition.
