Recently, some new convolution formulas extending the orthogonality of the Stirling numbers of the first and second kind were shown by algebraic techniques. The formulas involve sums of products of the two Stirling numbers wherein the inner arguments vary while differing by a prescribed amount and the outer arguments are fixed. Here, we provide combinatorial proofs of these formulas using direct enumeration and sign-changing involutions. Our arguments may be extended to establish generalizations of the foregoing results in terms of the r-Stirling numbers.