This phrase describes the first year or so after finishing your schooling. It can mean after high school, after college, or even after law school, medical school, business school, or grad school - whatever type of school a person goes to before completing their education and getting a job.
This phrase is usually used to talk about your first job situation after graduating. You can use the word "school" like in the example above, or you can specify which type of school:
I started my first business right out of college.