Here's the Luke 11 passage as a whole...
Luke 11:14-23
And he was casting out a demon, and it was mute; when the demon had gone out, the mute man spoke; and the crowds were amazed. But some of them said, "He casts out demons by Beelzebul, the ruler of the demons." Others, to test him, were demanding of Him a sign from heaven. But he knew their thoughts and said to them, "Any kingdom divided against itself is laid waste; and a house divided against itself falls. "If Satan also is divided against himself, how will his kingdom stand? For you say that I cast out demons by Beelzebul. And if I by Beelzebul cast out demons, by whom do your sons cast them out? So they will be your judges. But if I cast out demons by the finger of God, then the kingdom of God has come upon you. When a strong man, fully armed, guards his own house, his possessions are undisturbed. But when someone stronger than he attacks him and overpowers him, he takes away from him all his armor on which he had relied and distributes his plunder. He who is not with me is against me; and he who does not gather with me, scatters."
Jesus has just cast a demon out of a man who'd been made mute by the demon, and now the man could speak and the crowds were amazed. Their amazement, however, is not simply that Jesus was able to cast out a demon but that he did so by the power of Beelzebub. Why should they be amazed someone has the power of Beelzebub? Why should they be amazed Jesus has the power of Beelzebub to cast out demons over which Beelzebub ruled? Since Beelzebub is king of the demons, he certainly has such power. There's nothing amazing about that power. What's amazing is that a man should have that power.
Jesus later asks by who do the sons of Israel cast out demons, implying both the sons of Israel and Jesus cast out demons by the power of someone other than Beelzebub. Jesus draws attention to the inconsistency in their thinking and reaction. Notice assumes, accepts, and asserts as a given their sons cast out demons by some power other than Beelzebub's - otherwise the audience must acknowledge their sons have the power of Beelzebub, too. It also implies their lack of faith: God couldn't possibly be behind Jesus' power; it must be due to Beelzebub. Beelzebub is, apparently, very powerful.
The problem is: Beelzebub is a myth! Beelzebub was a pagan god of Philistines, not the one true God, the God of Abraham, Isaac, and Jacob, not the only God that actually exists. Beelzebub is not Satan. In other words, to understand what the audience had just witnessed they appealed to the pagan cultures, not Tanakh or monotheist Judaism. The episode subtly betrays how far from God Jesus' audience had fallen.
Notice also Jesus explicitly states the kingdom of God had come upon them. Right then, right there in front of them, right before their very eyes the kingdom had come. Its witness was Jesus commanding demons and liberating a son of Abraham. His audience attributed it to a pagan god that didn't actually exist. They attributed the kingdom action to the lord of the flies.
Jesus responds by the analogy of the possession guarding strong man who is attacked and overpowered and then subsequently has his possessions taken away by someone stronger than the strong man. The word in the Greek for "plunder" or "spoils" is a word specifically for possessions taken from an enemy or foe. In ancient time a portion of a soldiers pay was what loot he could carry from his enemy once the enemy was defeated. The soldiers wage for fighting was meager, but what he might be able to carry off from a defeated adversary could make him wealthy.
So what does Satan possess that Jesus can take for his own?
The demon possessed..... slave to sin.
That last part is important because Satan and his other demons cannot possess a sinless person. Furthermore, Satan and the other demons are themselves also enslaved. The wages of sin is death (Rom. 6:23). That is just as true for the adversary and his fellow devils as it is from every human who disobeys God. The reality is the one owning the slave of sin who is dead in his or her transgressions is himself a slave of sin who is also dead in his transgression. He's not actually a strong man. He is neither strong, nor human. Jesus came to set the captives free from sin and undo the works of the devil. The episode described in Luke 11 is Jesus doing exactly what he came to do. Jesus is the only actual strong man in the episode, but his audience does not understand that. They think Beelzebub really exists and they think Jesus is acting within Beelzebub's power and not his own power. By describing how an even stronger person is necessary to overpower a lesser strong man he is imply he is the stronger man, a man stronger than the demon he cast out, stronger than the mythical Beelzebub his audience imagines to be at work. It never occurs to them a Spirit greater than the possessing demon or that demon's lord, Beelzebub, could be the impetus for what they'd just witnessed.
