Nations with ancient cultures (and others as well,) are insisting that their cultural artifacts be returned to them for two main reasons. First, they want to have control of the relics of their own past. Second, they are offended by the idea that other countries took control of those relics, typically at a time when they had colonized the countries by force.
Countries naturally want to have control over the relics of their own past. They want to be able to display their pasts proudly. They want to celebrate their history and the things that they created. By getting their artifacts back, they can do these things.
Countries also resent the way in which their cultural artifacts came to reside in other countries. This typically happened when powerful countries dominated weak countries. For example, English scholars might have removed artifacts from Egypt at a time when Egypt was under British control. This seems unfair to the people of the countries from which the artifacts came. It seems unfair to them that countries could take artifacts, essentially by force, and could now benefit from those artifacts.
For these reasons, many countries want cultural artifacts of theirs returned.