Complex sentence: The "k"-vectors have degree "k", meaning that they are sums of products of "k" vectors, and when elements of different degrees are multiplied, the degrees add (like multiplication of polynomials).
Split into simpler sentences: The "k"-vectors have degree "k", meaning that they are sums of products of "k" vectors. When elements of different degrees are multiplied, the degrees add (like multiplication of polynomials).