Other forms: weathered the storm; weathering the storm; weathers the storm
When you weather the storm, you make it through something particularly difficult. The difficulty could be physical danger, or it could be emotional, political, or professional.
It's rare nowadays to use the term weather as a verb, and we do often use it to mean "wear out." But it was used before to mean "get through a period of bad weather." You could weather out the winter, or weather a cold night. Someone who has weathered the storm has gotten through that period of bad times, and, like a ship coming safely into harbor, is now safe and sound.