In the NMF factorization, the parameter k (noted r in most literature) is the rank of the approximation of V and is chosen such that k<min(m,n). The choice of the parameter determines the representation of your data V in an over-complete basis composed of the columns of W; the wi , i=1,2,⋯,k . The results is that the ranks of matrices W and H have an upper bound of k and the product WH is a low rank approximation of V; also k at most. Hence the choice of k<min(m,n) should constitute a dimensionality reduction where V can be generated/spanned from the aforementioned basis vectors.
Further details can be found in chapter 6 of this book by S. Theodoridis and K. Koutroumbas.
After minimization of your chosen cost function with respect to W and H, the optimal choice of k, (chosen empirically by working with different feature sub-spaces) should give V∗, an approximation of V, with features representative of your initial data matrix V.
Working with different feature sub-spaces in the sense that, k the number of columns in W, is the number of basis vectors in the NMF sub-space. And empirically working with different values of k is tantamount to working with different dimensionality-reduced feature spaces.