“words”是指”话语，字句”，”weigh” 除了称重外，还有“斟酌”之意。“weigh one's words”意思就是斟酌字句。

【例句】
Having a meeting with your boss each time, once the boss asks you to have a talk, be careful, you have to weigh your words, otherwise the boss would call names.
每次同老板开会，一旦老板要求你发言，小心了，你必须斟酌字句，否则老板就会骂人。

In order to write a good article, you have to weigh your words.
为了能写出一篇好文章，你不得不斟酌字句。