While the training mode has been available in most (if not all) of the Street Fighter games, this stage is designed specifically for the purposes of practicing. Trials and Training Mode (in the Street Fighter IV games only) automatically take place here.
In all of its appearances so far, the Training Stage is designed to minimize distraction, which would be caused by the background and other interactive elements within a stage. As a result, the room is flat and mostly featureless, save for the inclusion of evenly marked lines. The blue squares on the floor mark the 'plane' where the gameplay takes place, and blue squares are placed around the red lines that divide the stages into equal halves.
These lines can be used to measure distances for attacks, and allow players to study the spacing that may be required for a combo. In Street Fighter X Tekken, the marking of Pandora can be seen on the wall of the stage opposite the "fourth" wall.
In the Street Fighter IV games, Dan serves as the training dummy for trials, wearing his training gear. In Street Fighter X Tekken, Ryu is the training dummy for all Tekken characters in trials, while Kazuya is the training dummy for all Street Fighter characters in trials (and, by extension, the game's opening tutorial).
The Street Fighter V iteration, The Grid, is revealed to be part of a Shadaloo testing facility - more specifically, F.A.N.G's lab, which he uses to test kidnapped fighters and have them battle against simulations until their death. Alex's story mode has him destroying the lab after being forced into fighting simulations, forcing F.A.N.G to set him loose.