What does Luke 18:15 mean?

Because Luke arranged much of this section by theme, rather than by time, we're not given much context. We don't know if Jesus is traveling or teaching or performing miracles. The text doesn't say if the children need healing, as in Luke 6:19, or if it's the night before the Day of Atonement when elders bless children. Apparently, none of those details are necessary for Luke's purposes.

The conflict comes when parents want Jesus' attention for their children, but the disciples won't allow it. In fact, they actively rebuke the parents for contemplating such a thing. Either they don't want Jesus to be interrupted, or they think children are beneath Him.

It's likely the latter. Luke says Jesus countermands the disciples' order (Luke 18:16). Mark says Jesus is "indignant" (Mark 10:14). Jesus believes that welcoming children into the kingdom of God is well worth whatever interruption is incurred.

Translated into English, the Greek word brephē—here translated as "infants"—can generate confusion. In Luke 18:16–17, Jesus talks about these "children"—using a different Greek word—receiving the kingdom of God, which "infants" can't do. But in 2 Timothy 3:15, Paul uses the same Greek root word to refer to Timothy when he was old enough to listen to the Scriptures. Rather than meaning a baby or an extremely young child, the Greek term is a more generic term for those who are young.