Television seasons are written by writing staffs, not a single writer. Shows therefore employ — and, accordingly, need to hire — entire staffs of writers. This is true of new shows as well as returning shows, as often there is turnover from season to season.
Staffing season is the period when TV shows hire their writing staffs for the next season, after networks decide which pilots and returning shows will be picked up for the season, and before the new season’s writers rooms begin.
From roughly mid-April to late-May, showrunners along with network, production company, and studio executives read specs and take meetings with writers who are hoping to be staffed. For those roughly-six weeks, there is a mad dash from all sides as everyone tries to find their perfect prom dates.
A few things to note:
Higher level writers are hired first.
More and more, cable is operating on a different schedule than the traditional network schedule, and therefore may hire partial or whole staffs outside of staffing season.
Mid-season pickups are often staffed outside of staffing season, as their rooms start later.
A show that is already on air and knows that there will be a next season may staff earlier.
If a show fires a writer, they may try to hire a replacement mid-season, outside of staffing season.