Basically, it looks like this:

(Image rendered in POV-Ray by the author, using a recursively constructed mesh, some area lights and lots of anti-aliasing.)
In the picture, the blue square on the $x$-$y$ plane represents the unit square $[0,1]^2$, and the yellow shape is the graph $z = x \oplus y$ over this square, where $\oplus$ denotes bitwise $\rm xor$.
Note that this graph is discontinuous at a dense subset of the plane. In the 3D rendering above, no attempt has been made to accurately portray the precise value of $x \oplus y$ at the points of discontinuity, and indeed, it is not generally uniquely defined. That is because the discontinuities occur at points where $x$ or $y$ is a dyadic fraction, and therefore has two possible binary expansions (e.g. $\frac12 = 0.100000\dots_2 = 0.011111\dots_2$).
As can be seen from the picture, the graph is self-similar, in the sense that the full graph over $[0,1]^2$ consists of four scaled-down and translated copies of itself. Indeed, this self-similarity is evident from the properties of the $\oplus$ operation, namely that:
- $\displaystyle \frac x2 \oplus \frac y2 = \frac{x \oplus y}2$, and
- $\displaystyle x \oplus \left(y \oplus \frac12\right) = \left(x \oplus \frac12\right) \oplus y = (x \oplus y) \oplus \frac12$.
The first property implies that the graph of $x \oplus y$ over the bottom left quarter $[0,1/2]^2$ of the square $[0,1]^2$ is a scaled-down copy of the full graph, while the second property implies that the graphs of $x \oplus y$ in the other quarters are identical to the first quarter, except that the lower right and upper left ones are translated up by $\frac12$.
The resulting fractal shape is also known as the Tetrix or the Sierpinski tetrahedron, and is a 3D analogue of the 2-dimensional Sierpinski triangle, which is also closely linked with the $\rm xor$ operation — one way to construct approximations of the Sierpinski triangle is to compute $2^n$ rows of Pascal's triangle using integer addition modulo $2$, which is equivalent to logical $\rm xor$.
It may be surprising to observe that this fully 3-dimensional fractal shape is indeed (at least approximately, ignoring the pesky multivaluedness issues at the discontinuities) the graph of a function in the $x$-$y$ plane. Yet, when viewed from above, each of the four sub-tetrahedra indeed precisely covers one quarter of the full unit square (and each of the 16 sub-sub-tetrahedra covers one quarter of a quarter, and so on...).