"With" is often used for associations. That is, when a person is connected to an organization (business, charity, school, etc.) we can say they are "with" the organization.
It is a more general term that can replace "work for," "volunteer for" or "participate in." But it typically is used to show that there is a close relationship and that the person has permission to represent the organization.
For example, "I'm with the political campaign of John Smith." A volunteer who doesn't get paid might say this as they go door to door asking people to vote for Mr. Smith.
They could also say, "I volunteer for the political campaign of John Smith." It would be the same meaning, but this sentence is more specific because it is clear that the person is a volunteer and not an employee.