What does John 10:27 mean?

This continues Jesus' repetition of an analogy He made several months prior, after healing a man born blind (John 9). There, He pointed out that sheep only recognize the voice of their own shepherd; those who don't listen to the witness of Jesus are proving they're not part of His "flock" (John 10:1–6). In another analogy, Jesus stated that He was the only means of salvation (John 10:7–9), separating all people into two basic groups: saved and unsaved. Those who refuse to come to Christ are, by definition, part of the group mastered by Satan (John 8:42–47).

Jesus makes this statement under dire circumstances. His critics have trapped Him in a corner of the temple (John 10:24), daring Him to repeat His claims, and apparently prepared for violence (John 10:31). In typical fashion, Jesus not only responds with brave truth, but He also continues (John 10:28–29), culminating in a statement that seems almost deliberately intended to enrage His critics (John 10:30).