Why when we look through a telescope in space, do the billions of stars not block our view from seeing further? I know that we are able to "see back into time" by looking very far away through telescopes such as the Hubble telescope, but my question is, wouldn't you run into at least a couple of stars that were blocking your vision when you were trying to look even farther? 

Imagine a big Boulder in front of another smaller Boulder  and if we are looking from the side of the larger Boulder, we wouldn't be able to see the smaller one because the bigger one was in the way. Hopefully that simplifies my question. If it is as simple as that, wouldn't we miss stars and not be able to view them?
OK, Imagine the stars at distance x block an area of the sky. At a distance of 2x there should be four times as many stars, but they would seem four times smaller in terms of area covered. Thus, the blocking grows linearly until a significant part of the sky is blocked. The stars within 20 light years (excluding the Sun), blocks approximately $4.3 \cdot 10^{-16}$ of the sky. Even if we scale that up to the radius of the universe, 45 billion light years, only about $2 \cdot 10^{-6}$ of the sky is covered.

Even that tiny number is an overestimate, as we live in an extremely star-dense part of the universe, our galaxy.

Space is simply too big.