Well, the easiest thing is to connect similarity and causality. This is the foundation of so called sympathetic magic. If your voodoo doll is similar to you (e.g. by having some of your hair), actions I perform on it, will also happen in you.
Temporality and causality are related in a similar way. I used the example of 'since'. The original meaning is 'from a time', but because when one thing precedes another, it often causes it (in the same way that voodoo dolls do), the meaning can easily shift into one of causality. 'Since we are here, let's have some fun.'.
And the same process applies to the temporal meaning of 'as' developing from similarity.
How did 'as' acquire that temporal meaning? It was used to describe manner in which events happen. 'He ran as if he was possessed.' Ie. 'he was similar to someone possessed.' Once you have usage in that context, it becomes more plausible to say 'As he ran by the graveyard everyday, he saw a ghost.'
So the chain from similarity to temporality is established.
And then it's just a short step to causality. 'As he ran by the graveyard every day, he was eventually possessed by the ghost.' (Running came first, then possession. Thus running caused possession.)
I guess, the problem is with the word 'see'. It may sound implausible but analogous processes happen all over languages all the time and if you look at enough of them, you sort of 'naturally' begin to see these relationships.
Also, they may not be as obvious as seeing something but they are clearly how speakers 'integrate' or 'blend' meanings without trying (something the blending theory describes really plausibly. However, there is also variation among speakers and some resist these semantic extensions.
So if you don't find any of this obvious, just accept it happens. Just like most people accept gravity just happens without understanding any of the mathematical models behind it. In a way, this was not obvious a priori, it was established through empirical observation. To continue the physics analogy, we are at the level of Newton (or perhaps Galileo) as to how and why (we know what and can describe it), we are still waiting for an Einstein.