"Astronomers have concluded that the universe began some 12 to 15 billion years ago. That means we can only see the part of it that lies within 12 to 15 billion light-years from us."

This is false, because it ignores the metric expansion of spacetime. We can see radiation from objects that are now ~46.5 billion light-years from Earth.

This isn't picking nits. The explanation given here makes it sound like the Universe is a Euclidean volume that got instantaneously filled with stars, at roughly their present density, and that this density is low enough that light from distant stars hasn't reached us yet. If this were true, it wouldn't only explain why the sky is dark now, but predict that it has always been dark. We know this is false: for the first 380,000 years after the Big Bang, the Universe was filled with hydrogen plasma and completely opaque to radiation, which we can still see in the form of CMBR. As spacetime expands, energy density decreases, space becomes less light-emitting and more transparent, and light already in transit becomes red-shifted. Our field of vision is still technically flooded with light, as it always has been, but by now the red-shift is so extreme that the light is in the radio spectrum.