What does Mark 6:33 mean?

The Gospel writers, like most ancients, are not as specific about times and places as modern readers would prefer. It's likely Jesus and the Twelve left from Capernaum. This is where Peter, Andrew, James, and John live, and where Jesus makes his ministry center. Luke 9:10 says Jesus took them to Bethsaida. This agrees with Matthew 14:22, as Bethsaida is on the other side of the Jordan River, which both feeds and drains the Sea of Galilee.

The plain of Bethsaida easily fits the description of a "desolate" place (Mark 6:32), since the Greek word erēmon refers to an area which is uncultivated, lonely, or secluded. This is why the same place can also be said to contain green grass (Mark 6:39). However, Mark 6:45, which describes the return home, seems to infer Bethsaida is "on the other side" of the desolate place.

Most likely, Jesus meant for the Twelve to go from the plain to Bethsaida to get the boat, then continue to the other side of the Sea toward Capernaum (John 6:16–17).

Capernaum is only about six miles away from Bethsaida, so it's reasonable that some of the people can quickly cover the distance and arrive before the Twelve have a chance to row to Bethsaida, moor the boat, and walk to the plain. It's also possible that "got there ahead of them" means "anticipated where they were going." The parallel passages in Matthew 14:13, Luke 9:11, and John 6:2 say the people "followed" Jesus. It's likely that some men ran quickly and made it there first while others followed behind.

Jesus has been a victim of His own success since the day He freed the demon-possessed man in the synagogue and healed Peter's mother-in-law (Mark 1:21–34). Now the crowd recognizes individual members of the Twelve as well. The Twelve are looking forward to some rest after a busy missions trip, but the success of that very mission leaves the people wanting more.