This is a term used in baseball to refer to a series of games played at a team's home field. It may involve one opponent or many. Usually a home stand must involve at least two or three games. The phrase comes from the notion that the home team must "stand up" to the opponent, make a stand to win the game. Since there are so many regular season games, every team has multiple home stands per season. This allows the team to build momentum, and perhaps gain an advantage since they are playing at home, before they go out on the road.
The opposite of a home stand would be a road trip, in which a team plays many games away from their home stadium. In Major League Baseball, teams may play as many as seven or eight games at home before traveling on the road again. The phrase could apply to other sports as well, so long as they play multiple games in a row at home.