Self-similarity means that recursive algorithms can be used to simplify the process of creating complex patterns.
Without self-similarity and recursion, the coding required to define the structure of a tree or that of our nervous and circulatory systems would be incredibly complex and it becomes impossible to conceive of how these could evolve via random processes. But with it they boil down to simple branching instructions repeated a few times in a feedback loop.
The macroscopic is a manifestation of the microscopic and the same patterns appear at every level of scale. As the saying goes it's "turtles all the way down".