It means to be watchful or vigilant.
*The police were watching the proceedings, Argus-eyed. *The students found it difficult to get past the Argus-eyed watchman.
Origin of the usage
Argus was a giant who had fifty pairs of eyes. Juno, wife of Zeus, asked him to keep an eye on her heifer (young cow) Io. The vigilant Argus stood guard, and when he felt drowsy, he allowed only two of his eyes to fall asleep. The remaining forty-nine pairs of eyes were focussed on Io. Unfortunately for the giant, the messenger of the gods, Mercury, became interested in the heifer and decided to steal it. In order to achieve this, he began playing his lyre. The music was so soothing that Argus fell asleep – all one hundred eyes closed. Mercury drew his sword and promptly chopped off the giant’s head. When Juno saw what had happened, she removed the eyes from the head of the giant and placed them all on the tail of a peacock. I guess this explains why we talk about the eye of a peacock’s feather.