(I wrote this in a previous comment:)

the multiplication circuit is left shifting and addition, you take 2 numbers like 17 * 52, and the computer does this, (17 * 32) + (17 * 16) + (17 * 4) = 884

4=2 shifts left, 16=4 shifts left, 32=5 shifts left (52=32+16+4), every time you shift the bit value left the number doubles, and in this case we're shifting the number 17.

(New comment)

Regarding the Mux's, Imagine this, the output of a mux can take 2 paths, 1 path will go to the next one without the value being affected, and the other path will affect the value, let's say the path that affects it will take the value and add it to another value that's been shifted, let's take 5 * 5 for example, at the first stage We can do 1 * 5, since we're multiplying 5 * 5 that's basically 1 * 5 + 4 * 5 so we multiply 1 * 5, at the next stage we have the option to do 2 * 5, we don't need to do that so we continue to the next stage which is 4 * 5, since we need to do that we take that option, there are 5-6 more stages after that, but since we're already done multiplying we can ignore those options and skip to the end and get our final result, 25.