What do m and s even represent?

past or present denominators, or defaults

If the value before the most recent left choice was l/m and the value before the most recent right choice was r/s then the new value will be (l+r) / (m+s).

That line, paraphrased / decoded / whatever:

(l+r)/(m+s) is each new fraction (next iteration), before substitution and simplification. Let l/m represent whatever the fraction was, just before you most recently parsed an L choice. Let r/s represent whatever the fraction was, just before you most recently parsed an R choice.

This is how values of l, r, m, s are obtained most often. But l = 1, r = 0, m = 0, s = 1 are defaults, used when there is no previously parsed L or R (or neither, as shown in the way that the initial 1/1, corresponding to the empty string, is formed entirely out of them).

HTH

Same here, I took my notebook and wrote the numbers, the fractions, tried to find a pattern but nothing. I'm not quite following the explanation, this is a problem that most likely would be solved in lets say 10-15 lines maybe, but man I can't even find the logic. If someone can explain it I would appreciate it a lot.

what the hell

This seems like it shouldn't be terribly difficult to implement, but, I can't make any sense of what is actually being asked. I'm trying to follow the "let's take a walk" examples as well, but to no avail.

If there have been no left choices, l = 1. Why isn't it 0 since left has been chosen 0 times so far?

If there have been no right choices, r = 0. Okay maybe that makes sense, as right has been chosen 0 times. But then why is s = 1 instead of 0? Or, why is m = 0 instead of 1? What do m and s even represent?

Is the following test case erroneous?

No the test is correct. You can notice that 5096 guys passed the Java kata.

Since they say that the word "and" may or may not come in between numbers, couldn't you just replace all the "and"s with before digging in?