Why is fat sticking out of corner of my left eye? (photos)

It's hard to tell definitively from the photos, but that appears to be prolapsed orbital fat, or fat which normally surrounds your eye as protective covering that has migrated outside its usual position. This is a slightly odd location as these are more commonly seen at the superior lateral (higher and outside) portion of the eye/eyelid junction. These can typically be removed if they become problematic (ie eye irritation, partial vision obstruction, etc.).
Hope this helps!Kyle Kimura, MD Facial Plastic and Reconstructive Surgery

You need to be seen, but this looks like a prolapse of fat that is normally behind the eyeball and now presents itself under the conjunctiva which is the layer that covers the eyeball. It normally doesn’t cause any problems, but it can be a cosmetic issue that may require removal. I stress again that you need to be seen to confirm my statement as you can’t make a diagnosis with absolute certainty on a photograph. 

Your photos are consistent with a common condition that comes with aging -- especially in people who have shallow orbits and lax eyelids. There is some orbital fat that can bulge forward and be noticed as your photos seem to show. But only during a complete examination with an expert can a definite diagnosis and treatment plan be made. You need an in-person consultation with a board-certified, fellowship-trained Oculofacial Plastic Surgeon with extensive experience in orbital and  eyelid surgery.  Many patients travel great distances to see me for this sort of expertise. You must avoid any doctors that perform orbital or eyelid surgery part time, they cannot help you. Bulging orbital fat is not a medically covered procedure -- it is considered cosmetic surgery by insurance companies. I have performed this procedure many times with beautiful results.
Best of luck to you. 
Damon B. Chandler, MD
Harvard-Penn Trained Oculofacial Plastic Surgeon

Unfortunately you have not given us any other information.  Is this natural or the result of a procedure? Does it cause any problems at the point or are you just curious?

Thank you for sharing your case and photos.  This is a common condition where the fat that is normally behind the eye starts to stick out.  It can be removed during a simple outpatient surgery.  Schedule a consultation with a board certified oculoplastic surgeon to discuss your condition and make sure your eyes are otherwise healthy.  Good luck!