What does "their worm and their thirst will never be quenched" mean?

Your question refers to a Bible description of hell. Three verses in scripture (Mark 9:44Where their worm dieth not, and the fire is not quenched.
See All..., 46, 48) are identical. They say of hell, "Where their worm dieth not, and the fire is not quenched." This description is taken from this Old Testament passage: Isaiah 66:24And they shall go forth, and look upon the carcases of the men that have transgressed against me: for their worm shall not die, neither shall their fire be quenched; and they shall be an abhorring unto all flesh.
See All... - "And they shall go forth, and look upon the carcases of the men that have transgressed against me: for their worm shall not die, neither shall their fire be quenched; and they shall be an abhorring unto all flesh."

Notice first that the reference to the worm is a personal reference. It is "their worm." Therefore, it is much more than a reference to the worms in the trash dump in Valley of Gehenna as some teach. It is a description of the people in hell. It is "their worm" that does not die. This should not surprise us since man is compared to a worm in his lowest state.

Job 25:6How much less man, that is a worm? and the son of man, which is a worm?
See All... How much less man, that is a worm? and the son of man, which is a worm?

Psalm 22:6But I am a worm, and no man; a reproach of men, and despised of the people.
See All... But I am a worm, and no man; a reproach of men, and despised of the people.

The second reference (Psalm 22:6But I am a worm, and no man; a reproach of men, and despised of the people.
See All...) is a prophecy of the sufferings of Christ on the cross. He became a worm, an object of reproach and despite, when He took the sins of man upon Himself. In like manner, those who reject the payment of Christ will become as worms in hell. They will be reproached, despised, and rejected. The comparison is clear: either Christ became a worm for us on the cross or we will exist as a worm in hell eternally. Just as Christ ceased to look like a man when He suffered for our sins on the cross ("his visage was so marred more than any man, and his form more than the sons of men" Isaiah 52:14As many were astonied at thee; his visage was so marred more than any man, and his form more than the sons of men:
See All...); so, those who reject the payment of Christ for their sins will suffer eternally for their own sins in a form that does not resemble a man.

Now, I will allow the Lord to make the final application of the text. I am not claiming that all human shape is lost. I do not know. However, the comparison to a worm is powerful. We should certainly take God's offer of salvation seriously.

You also mention their thirst not being quenched. In the scriptural statements, it is the fire that is not quenched. However, this certainly leads to thirst. In the story of the rich man in hell, the rich man asks Abraham to send Lazarus, "that he may dip the tip of his finger in water, and cool my tongue; for I am tormented in this flame." Here, the flame and thirst are connected. The statement about the fire never being quenched simply means that the fires of hell will never go out.

This one simple description of hell -- "Where their worm dieth not, and the fire is not quenched" -- should make all of us want to make sure that this is not our eternal destination. But it does not need to be. Jesus Christ gave Himself on the cross and arose from the dead so that we might have our sins forgiven and live eternally with Him. I pray that you have truly trusted in Him. If not, it is not too late.