This is something I have been struggling with for long time. I don't see why, I don't understand how でしょう and だろう convey the meaning of "probably" as in sentences such as 雨が降るでしょう。It simply doesn't click in my head. Perhaps an etymological explanation could makes sense of this. That would be really appreciated.
First, the -う at the end of だろう and でしょう is an auxiliary verb that used to have a broader function. Now it is mainly used to express intent (〜(よ)うと思う) and make propositions (行こう or 食べよう), but up until the Edo period it also expressed uncertainty and conjecture. You can perhaps see how uncertainty, intent and proposition are conceptually related as they all rely on things not yet being determined or concluded.
-う is actually a sound changed form of the auxiliary verb -む. You can find vestiges of -む and its contracted form, -ん, in set phrases like 〜といわんばかりに (as if to say ~). It also appears as the む in the Iroha poem.
Second, the だろ- of だろう is what you get when you conjugate the copula to attach -う and then subject it to some sound changes. The でしょ- of でしょう is the same thing plus a bonus auxiliary verb (-す) thrown in to indicate politeness.
All together, だろう expresses uncertainty (indicated by -う) about the existence (indicated by the copula) of whatever precedes it. And でしょう does the same thing more politely.
Both of those roughly sound like the vocational form, which ends in おう, ろう, etc. So those may be old derivations of で + しよう and だXろう, where X is some verb stem. You can also try and look up the Japanese word for etymology, and them search that plus the phrases you'd like, and try to decipher the text you find. But, those may be pretty old terms which don't have a clear origin.