While traditional air conditioning systems chill air and blow it into your space, a radiant system conditions the objects and organisms inside of the environment rather than just the air.
Radiant systems do this by facilitating a thermal energy transfer between the objects/organisms in the space and a radiant surface. Heat is extracted from the space via the radiant surface which absorbs heat and gets slightly warmer. As the radiant surface absorbs heat, the system continues to pump cold water through the radiant surface to continue extracting heat from the environment to provide radiant cooling.
In terms of the feeling, radiant cooling is most comparable to walking into a cave or parking garage where the walls are very cold. As soon as you enter the environment, you feel cooler, and this is the sensation of thermal energy leaving your body (heat extraction via a radiant surface)!