The spot of light keeping itself in place while the character leaves the scene is meant for the audience to infer one of the basic problems in Samuel Beckett's play Krapp's Last Tape.
The play is quite one-sided. We only get our glimpse of the action in one setting, with one character, and with only one storyline. The action, itself, comes from Krapp and the tapes that he listens to. However, the tapes cause an effect on Krapp which leads him to leave the scene several times in order to "pop" a bottle of something to ease his angst.
The spotlight mostly remains in the same place while the character leaves the scene in and out, demonstrating to the audience how Krapp does not want to face reality and, instead, removes himself from it by entering the dark spots of the stage (which would represent his actual life, where he has remained in the dark).
Conclusively, the spotlight represents Krapp's present life and how Krapp shifts in and out of reality. This is the overall reason why the spotlight changes. Interestingly, the main character shifts in and out of the spotlight, as well, in order to demonstrate how he, at times, accepts and denies the place in life in which he exists.