At the risk of bringing on a headache, let me take my best and (I promise) only shot at an explanation. There are only two things you have to know. The first is by design and the second you will have to simply accept.
1. Numbers are marked on slide rule slides at a distance equal to log of number
2. log(x) + log(y) = log(xy)
Here is how these two things combine to allow you to multiply.
Let’s say you want to multiply 2 and 3 on the slide rule.
You move the start of the slide bar to line up with 2 on the bottom. It’s distance from the start is log(2) by the first rule.
Now you move the cursor to 3 along the slide, which, again by the first rule, is an additional distance of log(3)
The distance of the cursor relative to the start is log(2) + log(3), which by the second rule equals log(2*3) = log(6).
When you look on the bottom to see where the cursor falls, it will be the number at a distance of log(6), which by the using the first rule in reverse has to be 6.