When a horned lizard feels threatened by a predator, its final defense response is to shoot blood from these flooded sinuses and out its eye sockets. As a result, the predator is often frightened and flees. The lizard also uses this mechanism to remove foreign particles from the surface of its eyes.
Paris recognizes Romeo as the man who did murder Tybalt (and Juliet by proxy) and assumes that Romeo has arrived with the intentions of desecrating the corpses of Romeo and/or Tybalt.