Why would the states dispute each other's claims to western territories?

I am assuming you are talking about the United States’ states in your question:
1. Population - more population meant more seats in the House of Representatives.
2. Resources - western territories had resources that were not found (or at least were not as abundant) in the east, like gold.  Access to those resources meant more income for the states.
3. Slavery – Southern States were concerned about a non-slave owning state majority so there was always a fight over whether the new states would allow slavery or not.

Early territories of the original colonies stretched far westward and eventually these became the first new post-Revolution states.

They didn't want the south to have more authority over the north and that was controlled by population and other things.