Is there a geometric interpretation for Reidemeister torsion?

Yes, there is a very nice geometric interpretation over any manifold $X$ satisfying $\chi(X)=0$, for a version of Reidemeister torsion spelled out in Turaev's paper "Euler structures, nonsingular vector fields, and torsions of Reidemeister type". Ian Agol's comment mentioned the Seiberg-Witten invariants, which equals this version of torsion in 3 dimensions. It turns out that both of these are equal to an $S^1$-valued Morse theoretic invariant $I(X)$ defined by Hutchings and Lee in their PhD work:

"Circle-valued Morse theory and Reidemeister torsion" (Geom. Top. Vol.3, 1999)

Roughly speaking, we equip $X$ with a suitable Morse function $f:X\to S^1$ for which all critical points and closed orbits of $-\nabla f$ are nondegenerate. Then $I(X)$ is defined by suitably counting the Morse-flowlines between critical points as well as the closed periodic orbits (remembering their periods). The invariant is independent of the choice of $f$ and it is identified with the Redeimester torsion (for a suitable homology orientation). When $\dim X=3$ and $b_1(X)>0$, this also recovers the Seiberg-Witten invariants. We can view $I(X)$ as the analog of the Gromov invariant in 4 dimensions which suitably counts $J$-holomorphic curves.

Here is a simple example which may make the "involvement of products of determinants of basis-change matrices" less opaque to you. When $f:X\to S^1$ is a fiber bundle with fiber $\Sigma$, the periodic flow of $-\nabla f$ defines a self-map $\varphi:\Sigma\to\Sigma$, and $I(X)$ effectively counts fixed points of $\varphi^k$ weighted by their Lefschetz-sign. The identification with torsion translates into the Lefschetz fixed-point theorem.

Have you looked at the relation with Whitehead torsion and simple homotopy theory? I forget the details and am no expert, but there is a lot of nice low dimensional geometric topology hidden in that theory if that is any help. Look at Milnor's paper, Bull. Amer. Math. Soc., 72 (3):(1966) 358–426. but also look at the Wikipedia articles on both types of torsion for a quick entry point to the ideas.