Why would someone socially reject you but then stare at you ?

why would you care, walk away from them and never look back.

Stare at you like throwing daggers?  Trying to make you a bit paranoid.  Blow if off as their problem & not yours.

A person perceived as “less than” or lacking because of class, reputation, social status, intellect, training, race, gender, identity, etc., is stared at as a passive-aggressive coping strategy that allows the person that stares to maintain their false feelings of superiority which in turn allows the person that stares to feel empowered and in control.

Your first update gives an insight. Its not flattering for you.

Maybe someone's spreading gossip about you.  However, the way to deal with a staring person is to say, "Excuse me.  Is there something that you need?"  

You hurt their feelings, so that is why they stare at you.  In their minds they are wondering why you would be so mean. Maybe they never saw that side of you, so you became an oddity to them.

maybe they would like that person disappear? the presence doesn't coincide with their rejection. or, more likely, they are simply incoherent and meddlesome.

They may not like your personality but are attracted to you by your looks or they may think your stuck up and a snob.

Because they actually want to **** you.