assume the claims

[kjc0123]

What's the meaning of "assuming the claims of American citizens" in the following sentence?

In 1819, in return for assuming the claims of American citizens to the amount of $5 million, the United States obtained from Spain both Florida and Spain's rights to the Oregon country in the Far West.

[mrs.pudding]

Any claims that American citizens had against Spain were taken over by the American Government.  In return, Spain gave America its and Florida's rights to Oregon.