You might want to look into space filling curves, which were first described by Peano and Hilbert in the late 1800's. These are continuous surjections from $[0,1]$ onto $[0,1]^2$ (and higher powers) but they are not bijections. However, they are visualizable to a certain extent. A quick Google search gave a lot of hits, in particular this one at Cut The Knot which has an illustrative java applet.

As for the existence of a bijection, you can derive it from the fact that $\aleph_0\cdot2 = \aleph_0$ and the usual exponent rules:
$$(2^{\aleph_0})^2 = 2^{\aleph_0\cdot2} = 2^{\aleph_0}$$
It is also easy to write an explicit bijection between Cantor space $\{0,1\}^{\mathbb{N}}$ (the space of infinite binary sequences) and its square by splitting the even and odd coordinates. This, together with a bijection between $\mathbb{R}$ and $\{0,1\}^{\mathbb{N}}$, gives what you want. Note that it is this last bijection which is harder to visualize. The reason is that $\mathbb{R}$ is connected while $\{0,1\}^{\mathbb{N}}$ is totally disconnected (with the product topology).