The 22-year-old was one of 13 players released by Aldershot Town in May when the club went into administration.
Brown has said that he is happy to link up with Cheltenham.
"It was a no-brainer for me [to join]," he told BBC Radio Gloucestershire.
"We are very unlucky not to be in League One, so if I can come here and maintain and add to the team then I'll be over the moon. I want to be at the right end of the division this season. Last season defensively, without patting myself and the other defenders on the back, we weren't at fault.
"We kept 14 clean sheets last season, 13 in the league, so we struggled for goals last season, and what I want to do next season is keep clean sheets."
Brown played 41 times for Aldershot last season.