Sate is made up of:

1. sa, an older relative of "that" demonstratives sō, sono, etc.; and
2. te, related to de and the verb ending te.

So its etymology is something like "that being so", and when sentences in which it appears are translated to English, it often ends up corresponding to phrases like "well, now" or "OK, then" or "let's see".

In modern texts sate is usually written in hiragana, さて, but kanji were assigned to it in the Golden Age. In fact, there were no less than three ways to write it without resorting to kana.

The simplest one to explain is 扠. The logic here is that one of the Chinese-derived pronunciations of 叉 is /sa/, and 手 (the full form of the part on the left there) is the standard kanji for the Japanese word /te/ (hand). /sa/ + /te/ = /sate/.

(扠 also has its own meaning, something like "pierce" or "pinch"; I'm not sure if Japanese orthographers overloaded this existing character with /sate/, or recreated it independently.)

The next simplest is 扨. This works on the same principle as 扠, except 叉 has been replaced by 刄, a variant of 刃. Since 刄 has no pronunciation anything like /sa/, I can only assume that the association is purely visual. I am also fairly sure that 扨 was invented in Japan.

The third, 偖, has a slightly more interesting story. It is a Chinese-derived character meaning "rip" or "tear" (?) that came to be used to write /sate/ by accident: the character 手+奢 was created here locally (奢 = /sa/), but succumbed before very long to the gravitational pull of the existing, superficially similar 偖.

Recall, if you will, that it is relatively rare to see Japanese and Chinese elements combined in a single word. In each of these three cases they are combined in a single character. Moreover, none of the elements used have any etymological relation at all to the meaning represented. Should any man doubt that perversity can be not only an art but also a craft, let him behold these breathtaking achievements.