THE PROTOTILE SET:
 

Our tiling will be made out of infinitely many tiles that are copies of these.  For convenience of notation, we will refer to the prototiles as being of types 0, 1, and 2 respectively.

THE SUBSTITUTION:
 

To make a self-similar tiling out of these tiles, we choose a substitution rule for each of the prototiles.  This rule can be thought of as an``inflate-and-subdivide" process.  Each tile is first expanded by some  set amount, then sliced up in some fashion into tiles which are copies of the original prototiles.

Here we chose to make our substitution rule by first inflating by a factor of 5 and then cutting up  the tiles as follows:   (we show the substitution on tiles 0, 1, and 2 respectively)
 

Note:  in these pictures, the tile boundaries are not drawn in.  Your eye will have to extrapolate them.  Each row and column has 5 tiles in it.
 

We can create larger and larger patches of tiles by repeating (or iterating) the substitution---starting, for instance, with a white tile (type 0),  substituting once, then inflating the whole  picture again by a factor of 5 and subdividing each tile according to its substitution rule.  Picture it:
 
 

    inflate-and-subdivide:       inflate-and-subdivide again:
 
 
 

After 2 iterations, the patch is a 25 by 25 array of colored tiles.  Look carefully, see the substitution for the white tile in all four corners?  No matter how many times you iterate the substitution on this patch, you will always have white tiles in all four corners.   We can substitute again a few times, obtaining patches which are 125 by 125, and then 625 by 625 arrays of colored tiles.  We show them below, but we'll have to draw them smaller in order to see the whole thing.  Theoretically, the tiles are the same size as before;  we're just looking at them from further away:

isn't that pretty?
 

THE SELF-SIMILAR TILING:

A self-similar tiling is one which is invariant under the substitution process.  That is, after you're done inflating-and-subdividing every tile in the tiling, you are left with the same tiling again.   The first thing to notice is that our self-similar tiling will have to be an infinite one, because the substituted tiling must be 5 times a large as the original, yet be the same as the original.  And that only works if the original is infinitely large.

One way to get a self-similar tiling for our substitution is to pick a 2 by 2 array of tiles and place it with the center at the origin.   Use the substitution rule ad infinitum, inflating outwards from the origin.    You'll notice that each new iteration leaves the previous one intact and just adds more to the outside.  Here is a sampling of a few iterations:

    after one iteration the center is the same: 
after one more iteration the center is the same again:

We continue substituting forever until we have covered the entire plane, and we have created a self-similar tiling!