What does Luke 2:16 mean?

These men are local shepherds, who moments before were visited by a large host of angels. The angels proclaimed the recent birth of Jesus. That included explaining where and how Jesus and His family could be found (Luke 2:8–14). Given how spectacular the announcement was, it's not surprising the men are in a hurry.

Scripture does not explicitly say why this news was given to common herdsmen, rather than to some important ruler or politician. However, the choice fits with God's use of "ordinary" people to accomplish His will. It also emphasizes the humble circumstances of Jesus' earthly life, and the disinterest of His era's leaders.

As expected, the men find Mary, Joseph, and Jesus, in exactly the state depicted by the angels (Luke 2:4–7). Bethlehem was never a large community, so there would not have been many newborns to see, let alone any resting in a feeding trough. When the shepherds arrive, they will explain the vision that brought them to witness Christ's presence in the world (Luke 2:17).

Many cultures celebrate Christmas, in part, by setting out miniature figures representing the moment depicted in this verse. These "nativity" scenes usually include shepherds and Jesus' family. Many include the wise men described in Matthew's writing (Matthew 2:9–11). They almost always feature animals such as sheep and cattle. Those details are traditional, but not all of them are actually given in Scripture. The wise men, most certainly, did not meet Jesus until well after His birth.