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 $aleph_0cdot2 = aleph_0$ and the usual exponent rules:
$$(2^{aleph_0})^2 = 2^{aleph_0cdot2} = 2^{aleph_0}$$
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).
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