When Cooper lets his pod go into the singularity it's an act of sacrifice, with him accepting his fate in an attempt to save humanity. But it turns out it isn't the end; ejecting from the pod he finds himself in an endless landscape of strings of light. These provide him the ability to see his daughter's bedroom at an infinite number of points in time. In reality we've no idea what happens in a black hole and this is where the film fully steps into the fi part of sci-fi; Cooper (and TARS) are witnessing a three dimensional representation of a higher dimension. It's unclear whether it's the normal effects of a singularity or meddling by "them", although the latter is suggested in film, making this experience even more similar to 2001's Stargate sequence than it already is. Imagine how a wormhole bends the paper to put two points in three dimensional space that are lightyears apart next to each other and transfer that into the fourth dimension, time. He is able to jump to any time period, seeing various earlier moments in the film. He's able to lightly communicate with Murph by moving books and the like, becoming the ghost that she believed in growing up. Initially he believes he can somehow alter the future, but realising time is an unbreakable loop he instead puts his personal demons aside and uses it to influence the wider events on Earth (giving his earlier self the co-ordinates of the NASA facility and an older Murph the key to the gravity problem). Very fitting with the previous survival instincts talk.
