To weigh your words means to chose your words carefully before you say them.
Basically when you weigh your words you think about what you're going to say before you say them.
To weigh your words means to carefully think about everything you are going to say before you say it:
A good example of weigh your words is "He gave evidence to the court, weighing each word as he spoke"
You want to think carefully about the words you are going to say before you say them in court because the wrong words could get you into trouble.
It's never a good idea to lie in court because you can get into trouble and it will hurt your case much more.
So always weigh your words before you say them which means to simply choose the words carefully before you say them.
Another meaning could be if you don't have nothing nice to say or good to say to someone then don't say them at all.
That would be a way to weigh your words before you say them.