Why do I feel sad when I see people eating alone?
Because you're projecting how you would feel if eating alone... sad.
Your feeling, however, may not be at all in accord with what the lone diner is feeling. Introverts in particular are quite happy eating by themselves. In fact, certain kinds of company can spoil the whole occasion.
Most likely because you view dining as a social activity. That's fine, but for many — including me — it's an experience whose value lies primarily in the enjoyment of quality food, good service and the gracious ambiance of a good restaurant. I've dined alone in fine restaurants for decades with great satisfaction and no need or desire for company, which can often be a distraction from leisurely enjoyment of the dining experience.
You are telling yourself a story as to why the person is eating alone. You are mind reading, or attempting to guess the situation of a person you know nothing about.
That person might just want to have a bit of peace and quiet while eating. They need not be the social outcasts that you think they are.
Or perhaps it is you who have the problem? Maybe you are afraid to be alone? Maybe you grew up with a lot of siblings and have been surrounded with family members and were taught that you have to have people around you all of the time? Maybe you are afraid of what you will hear in your mind, when the sounds of the world shut off, and you are left with only yourself for company?