stagger : to arrange (things) in a series of different positions or times (Merriam-Webster)
On any curved track—a standard oval, for instance—the outside lanes are necessarily longer than the inside lanes. So that all runners end at the same finish after running the same distance, the starting points are staggered—runners in the inside lanes start farther back than runners in the outside lanes.
At the first turn the writer, in an inside lane, has "made up" some or all of the "stagger"—has reduced the distance which initially separated him from the runner in the next lane. Speaking of a "first turn" implies that there are more turns to come, so if the writer has actually drawn even with the other runner, or passed him, he has actually "gained ground" on his rival: traversed a larger proportion of the total distance.