Why does a fluid exert an upward force and not a solid?

The force/pressure exerts from every direction around the object. But the pressure get higher when you get deeper in the fluid (thanks to gravitation). So the net force is upward.

why? What inside a liquid exerts a force on its surroundings?

The atoms/molecules with their brownian motion hit everything around them and produce a net force so to speak. Surface tension acounts for a small part of preassure in every liquid and gravity plus depth presses on every layer beneath.