The best I can tell here is what it is saying. Given a finite group one can form the group algebra which is the set with pointwise scalar multplication/addition and the 'convolution' product where . Then, if denotes the set of all equivalency classes of irreps of one can select a representative (one can always put an inner product on a space for which any representation is unitary--so we can a priori assume that is a pre-Hilbert space and that the image of lives inside the unitary group ). If we then choose a basis (assume it's unitary) then for we can create the mappings where by (it turns out that is itself an irrep in the class ). We can then consider that functions for where is the complex number sitting in the entry of . Thus for every we have created elements of . I think then what it's stating is the fact that forms an (orthogonal) basis for .

That's how I interpret it, but a lot of the notation is foreign to me...so take that with a grain of salt.