The first example of an explanatory proof by induction that comes to mind is the solution to the following problem which originates (to my best knowledge) from Art Benjamin.
By a triomino I will mean an "L-shaped" union of 3 unit squares.
Claim: A $2^n \times 2^n$ grid of unit squares with one square removed can always be covered by triominos.
Proof:
Base case: if $n=1$ then we need to cover a $2 \times 2$ grid with one square removed by triominos. THat is, we need to cover a triomino by triominos. So we're good here.
Induction step: Assume that we can cover any $2^n$ by $2^n$ grid of squares with one square removed by triominos and suppose that we are presented with a $2^{n+1} \times 2^{n+1}$ grid of unit squares.
Separate this into a $2 \times 2$ grid of $2^n \times 2^n$ grids of squares one of which has one square removed. At the place where the four corners of these grids meet, place a triomino in such a way that it covers the corner square of each of the three $2^n \times 2^n$ grids without a square already having been removed.
Now what remains to be covered is the union of four $2^n \times 2^n$ grids of squares each with one square removed, so by the induction hypothesis we can cover it with triominos.