Why does the Mandelbrot set contain (slightly deformed) copies of itself?

I really like this question! I can't yet upvote, so I'll offer an answer instead. This is only a partial answer, as I don't fully understand this material myself.

Suppose that $f$ is a quadratic polynomial. Suppose that there is an integer $n$ and a domain $U \subset \mathbb{C}$ so that the $n$-th iterate, $f^n$, restricted to $U$ is a "quadratic-like map." Then we'll call $f$ renormalizable. (See Chapter 7 of McMullen's book "Complex dynamics and renormalization" for more precise definitions.) Now renormalization preserves the property of having a connected Julia set. Also the parameter space of "quadratic-like maps" is basically a copy of $\mathbb{C}$.

So, fix a quadratic polynomial $f$ and suppose that it renormalizes. Then, in the generic situation, all $g$ close to $f$ also renormalize using the same $n$ and almost the same $U$. This gives a map from a small region about $f$ to the space of quadratic-like maps. This gives a partial map from the small region to the Mandelbrot set and so explains the "local" self-similarity.

To sum up: all of the quadratic polynomials in a baby Mandelbrot set renormalize and all renormalize in essentially the same way. (I believe that there are issues as you approach the place where the baby is attached to the parent.) Thus renormalization explains why the baby Mandelbrot set appears.

It's slightly tricky for the Mandelbrot set, because this exists in parameter space. It's easier to think about the corresponding Julia sets - although the idea is similar.

The answer that I like is this: A Julia set is a "hall of mirrors". When you look at one, you are seeing a reflection of a reflection of a reflection... In fact, with the right software, you can animate this idea in real time. This also lets you see features of the set which aren't readily apparent just by looking at the usual escape-time colouring of it. But I digress...

When you think of the fractal as being a reflection of reflections of reflections, suddenly the idea that one small part of it might look like the whole thing seems... unsurprising.

If you want to really blow you mind, try this: Did you know that certain parts of the Mandelbrot set resemble particular Julia sets? I still haven't figured out exactly why yet...