What does John 13:19 mean?

Fear of the unknown is a powerful force in human life. When things seem to go wrong, a large measure of our emotion comes from uncertainty (Matthew 6:31–34). When "today" is different from what we expect, we lose confidence in what we assume about "tomorrow." To counter this, Jesus makes several statements specifically meant to strengthen His followers in times of hardship.

In this case, Jesus knows that Judas' impending betrayal will catch the other disciples totally off-guard (Matthew 26:14–16). Within the next few hours, they will see Judas lead a squad of soldiers and politicians directly to Jesus in the garden of Gethsemane. While the disciples will scatter in fear (Matthew 26:56), Jesus' words here are meant to ease their panic. It's exactly what He said was going to happen.

Similarly, Jesus will go on in this passage to warn Christians about persecution from the non-believing world (John 15:18—16:4). The point of that warning is meant to strengthen our faith. Troubles and setbacks are scary, but they are much less frightening when we realize they're happening just as predicted, and that prediction includes a good ending.