Most photographers who get assignments from national sports magazines start as sports photographers at local newspapers. These assignments are given by magazine's regional editors based on the location to pro photographers they know personally, or are recommended by other pro photographers, editors, league officials, teams, etc.
The AP or Associated Press is a popular news agency that gathers news and sports photos from hundreds of local newspapers and provides them to other member newspapers and magazines for re publication, Having your photos republished in national magazines will bring excellent attention to your skills. When you work larger events with national media coverage, you should do your best to talk and network with their shooters.
In case you haven't already figured it out, you need to know the names and job functions of the editorial staff and photographers of the top newspapers/magazines/teams in your city, find ways to connect with them in person and through social media, and show them that you are the best choice for these assignments.