Signification of the "dirty" variable ?

The main reason it is used it to call stage.invalidate() which triggers an Event.RENDER event. This is so that updates only happen once per frame. Event.RENDER is dispatched for all sprites that are be listening for it, so the dirty flag is used to keep track of whether the TileGrid should update or not. If another class called stage.invalidate, then dirty will be false and the TileGrid shouldn't update.

The reason we only want to update once per frame is, for example, when tweening the matrix values on TileGrid (tx, ty, a, b, c, d) we don't want to request and positioning tiles 6 times so we use Event.RENDER to ensure it's the end of the frame.

This works pretty well, I think, although it does cause some issues with events that are dispatched from the Event.RENDER handler. Nothing too bad, but that's why there's a MapEvent.RENDERED event that things like MarkerClip use to make sure the map is done moving before they update.