Short version: It isn't used for "cause" per se, but equivalence. That fourth definition could also be rendered dans la mesure où. (However, it may have spread from there to indicate cause.)
As Matthieu points out, the use of the word "cause" might not be quite right. It's tricky to pry apart causation, reasoning, and equation in any language. Let's look at some ideal specimens...
Here's an example of cause:
Jacques had to get up early, so he went to bed before nine.
Jacques' need to get up early directly led to his going to bed before nine.
(If we were reasoning this out, we could express it: P → Q ; P ; ∴ Q
Here's an example of reasoning:
When I got home the house was quiet, so Jacques went to bed before nine.
That is, I took the house's quietness as a sign of his action.
If we talk about causation here, the order is the reverse of what it appears to be: it is because he went to bed before nine that the house is quiet, not the other way round. The sentence structure disguises the fact that we are denying the consequent: If Jacques did not go to bed before nine, the house would not be quiet. The house is quiet. Therefore he did go to bed before nine.
(This argument expressed logically: ~P → ~Q ; Q ; ∴ P)
Now what about this one?
Jacques never tips his hat to a lady, so he is not a gentleman.
(Assuming that this is the key condition of gentlemanliness for our purposes...)
Let's try to read this as either of the above two species:
His failing to tip his hat leads to his not being a gentleman. This doesn't feel right, because one doesn't think of being a gentleman as a specific result of a specific cause. That is, it wasn't the case that he was a gentleman, and a second later he wasn't.
I see that he doesn't tip his hat and conclude that he's not a gentleman. Could make sense.
However, if we change the sentence a little, we get what we're looking for:
Jacques is not a gentleman, in that he never tips his hat to a lady.
or Jacques is not a gentleman; he never tips his hat to a lady.
This time there's no step from cause to effect nor from premise to conclusion. Rather, there's an equation. Jacques' ungentlemanliness consists in his not tipping his hat.
Knowing one, we could conclude the other, and vice versa. If he tipped his hat, he would be a gentleman. If he were a gentleman, he would tip his hat. This tells us that we aren't dealing with P → Q nor ~P → ~Q (since denying the antecedent shouldn't negate the consequent). Rather, this is P = Q.
This third sense is the one that d'autant que nominally carries. That is, it means dans la mesure où : insofar as, inasmuch as X is true, to that same degree Y is true.
Note that I said it nominally carries that sense. The reality is that these relationships are naturally very closely linked. Even that first example I gave above — "He had to get up early, so he went to bed before nine" — could have come out of someone's mouth as reasoning from premise to conclusion rather than (as an omniscient narrator?) stating cause and effect.
Because of this, it wouldn't be at all surprising if d'autant que had spread to be used for these other meanings. This answer merely explains how it got into the "cause" game in the first place.