This green section of the walkway, between Mount Safa and Marwa, is where men run (as part of the Umrah ritual) to symbolise the running of Hajra (a.s.). This occured when she was looking for water for her infant son Ismail (a.s.).
The house of Abbas (r.a.) was located just outside this space. He was the paternal uncle of the Prophet (ﷺ) who converted to Islam after the Battle of Badr.