Most of the new fancy techniques for learning multiplication and so on assume the student knows the tables.
For example, take a look at this lesson: http://www.homeschoolmath.net/teaching/md/distributive.php
Ignore the stupid rectangles, and ignore the part where it says "They actually use the distributive property, but we do not need to explain that to 4th grade students." (seriously, wtf)
A child who practices lots of sums, such that she knows how to add 420 and 56 without counting with their fingers or writing things down, will be able to learn this kind of multiplication with ease. Once they learn that and practice a lot, they will be able to generalize the method to multiply any two two-digit numbers.
Memorizing things is important because if you have to count with your fingers to calculate 3x3, you will never be able to calculate, for example, 23 squared in your head. But if you know the times tables your thought process might look like this:
20x20 = 400
23x20 = 400 + 60
23x23 = 460 + 23x3 = 460 + 69 = 529
(23 times 3 is "the hard part" where most kids who know all the theory (distributive property) but are out of practice (not enough rote memorization) will lose track)
A more advanced example is what engineers used to do a lot before they had calculators: They memorized log tables, so when they wanted to multiply big numbers they just added their logs together, because log(a x b) = log(a) + log(b).
An unrelated example is converting miles to kilometers. The official relationship is that a mile equals 1.609 kilometers, an ugly number that doesn't work well with mental arithmetic. But that number is kinda close to the golden ratio, don't you think? So if you are the kind of weirdo who memorizes the Fibonacci sequence, you can quickly calculate that 13 miles should be about 21 kilometers (and 21 miles is about 34km, and so on), because the ratio between a Fibonacci number and the next approaches phi.
