Don't worry doesn't mean don't care.
Jesus has just told them that God is the source of their gifts (12:22-34), that God's kingdom is what they should strive for (31) and what will be given to them (32).
Continuing with the image of kingdom then, Jesus discusses their role--they are to God the king as earthly servants are to earthly masters.
What worthy servants do is be ready at any moment for any need of the master. If the master tells you exactly what time he's going to show up, then you could take off work until that time came. But, if you don't know when he's going to get home, then you have to stay ready.
But, something unexpected occurs in Jesus' parable: when the master gets there, he will serve the meal to the servants.
Staying ready to open the door, keeping those lamps lit--only insiders can do that. (I've been reading Allen & Williamson'sPreaching the Gospel again.)
Lectio Divina: Luke 12:37; Psalm 33:20