You don't wait until someone tells you or asks you to do something; you do it on your own initiative.

你不会等到有人告诉你或要求你做某事；你是主动这样做的。

Then if you " do something off your own bat" -there's the bat-it means you do something on your own initiative.

那么如果你“do something off your bat”——这里有蝙蝠——意思是你主动做某事。

" I like you to go places and do things on your own initiative without me--you did much harder things for Rainy's publicity stunts" .

“我喜欢你在没有我的情况下自己去地方和做事——你为雨晴的宣传噱头做了更难的事情”。

Act on your own initiative, and noble France will appear again, almost such as our ancestors made her, and as our own eyes have seen her before the death of Louis XVI.

主动行动，高贵的法国将再次出现，几乎就像我们的祖先创造的那样，就像我们在路易十六死前亲眼所见的那样。