The short answer is human nature. The people of Jesus' time knew they needed help: Roman oppression, poverty, misery, religious confusion, division. Many false prophets and revolutionaries had arisen and failed to deliver. From their standpoint, any Jewish person would want proof that this new person promising help can deliver on his promise and not disappoint them or get them in worse trouble with the empire. That is why the people ask for a sign. The tricky question is why Jesus appears to refuse their seemingly honest, legitimate request.
Compare Luke 11:29-36 with a parallel passage in Matthew 6:19-24 that also speaks about eyes, seeing and light. When speaking of the Queen of the South and the people of Nineveh being more receptive to the words of the prophets than they are, he is shaming them, because those people, though they were pagan, yet recognized the light of God's wisdom and obeyed it. They acted in faith given much smaller demonstrations - signs - of God's power than Jesus had already performed.
19 “Do not store up for yourselves treasures on earth, where moths and
vermin destroy, and where thieves break in and steal. 20 But store up
for yourselves treasures in heaven, where moths and vermin do not
destroy, and where thieves do not break in and steal. 21 For where your
treasure is, there your heart will be also.
22 “The eye is the lamp of the body. If your eyes are healthy, your
whole body will be full of light. 23 But if your eyes are unhealthy,
your whole body will be full of darkness. If then the light within you
is darkness, how great is that darkness!
24 “No one can serve two masters. Either you will hate the one and love
the other, or you will be devoted to the one and despise the other.
You cannot serve both God and money.
The Matthew passage helps explain what is meant by "the light within". It is greed and materialism. It is a hunger for earthly treasures, not heavenly ones. Jesus is telling them that healthy eyes - eyes looking for heavenly treasures - already had enough evidence that he was a prophet and more. He was telling them that their problem was not his failure to present a sign, it was their spiritual blindness brought on by their sinful hearts.
Now returning to Luke 11:29-36
29 As the crowds increased, Jesus said, “This is a wicked generation.
It asks for a sign, but none will be given it except the sign of
Jonah. 30 For as Jonah was a sign to the Ninevites, so also will the
Son of Man be to this generation. 31 The Queen of the South will rise
at the judgment with the people of this generation and condemn them,
for she came from the ends of the earth to listen to Solomon’s wisdom;
and now something greater than Solomon is here. 32 The men of Nineveh
will stand up at the judgment with this generation and condemn it, for
they repented at the preaching of Jonah; and now something greater
than Jonah is here.
The Lamp of the Body 33 “No one lights a lamp and puts it in a place
where it will be hidden, or under a bowl. Instead they put it on its
stand, so that those who come in may see the light. 34 Your eye is the
lamp of your body. When your eyes are healthy, your whole body also
is full of light. But when they are unhealthy, your body also is
full of darkness. 35 See to it, then, that the light within you is not
darkness. 36 Therefore, if your whole body is full of light, and no
part of it dark, it will be just as full of light as when a lamp
shines its light on you.”
To the people demanding a sign, Jesus is saying that the ball is really in their court. It is not he who has to prove his authority, it is they who have to "see to it" that they purge the darkness from their hearts so that they can see what is already in front of them.
It is also interesting that Jesus is telling them to believe now based on a sign he will perform at a later date - namely his own resurrection. Recall Exodus 3, when Moses argued with God before the burning bush over his unworthiness.
12 And God said, “I will be with you. And this will be the sign to you
that it is I who have sent you: When you have brought the people out
of Egypt, you[b] will worship God on this mountain.”
The greatest of the signs that God promised to Moses was what would happen if he obeyed, and would only come after he obeyed. That is what Jesus is offering, and unlike Moses who obeyed despite his initial reluctance, they are determined to refuse.