meaning - Why can "bill" mean (almost) opposite things?

In general, this situation can arise for two main reasons:

(1) a word that originally meant a narrower concept diversifies to cover related concepts or specific connotations, and gradually over time the precise meanings of these once strongly related concepts can further diverge over time;

(2) what were once two separate words could merge due to sound changes or other phonological processes.

In this case, as mentioned in the answer above, it appears that effectively situation (1) has arisen: "bill" expanded from its original meaning of "official document" to cover various different specific connotations of "document" today, be it the "bill" put up by a "bill poster", the "bill" you pay for your meal, a "bill" of rights, a "dollar bill" etc.