Exactly what it did: claim supreme national authority over the people of the individual states. Meanwhile, the Southern states argued that since each state had voluntarily joined the union, it had the right to leave the union; because the Constitution did not expressly unite them as a single nation-state (below).
In 1776, each state was declared by its representatives to be a free and independent state, "with the full power to levy war, conclude peace, contract alliances, establish commerce, and do all the other things that free and independent states may of right do."1
And when these same representatives signed the Articles of Confederation in 1781, they each retained their state's sovereignty, freedom and independence.2
And when officially they won their independence in 1783 from Great Britain, the Treaty of Paris expressly recognized each state by name as being "free, sovereign and independent states." 3
So each state declared its existence in 1776 as a fully separate nation unto itself; and this was won by revolution in 1783.
Meanwhile, the Constitution was ratified by the "people" of each state; each by seceding their particular state from the Articles of Confederation. oblivious to the wishes of the other 12 states.4
The first of these was Deleware.
When the Deleware legislature presented the Constitution to the people of Deleware, they abdicated supreme legislative authority to the Deleware electorate: i.e. the citizen-voters of the state of Delaware.
This established the citizen-voters of each state, as the final authority of their own separate nation.5
This formally set precedent that each state could secede from compacts with other states, as could any sovereign nation-state.
The Constitution likewise did not unite the states together as a single new nation-state, as a simple reading shows. Unlike when the kingdoms of England and Scotland expressly united to form the single new kingdom of Great Britain in 1707;6 never did the United States ever expressly unite to form a single new state.
Instead, the Constitution-- just like the Declaration of Independence and the Articles of Confederation-- was a compact among separate nation-states.
Just like the Articles of Confederation had been: or like the EU or the UN today.
The difference, was that it was between the people of each ratifying nation-state; rather than the state governments.
Again, this began with the people of Delaware, who were the first to ratify the Constitution; and then the people of Pennsylvania ratified the Constitution.
But each state remained a separate nation, and its respective people remained the final authority over their respective nation-state.
Again, the Constition simply secured supreme legislative authority to the state's respective citizen-voters; and they in turn simply delegated power to their subordinates in the state and federal government.
The rest of the Constitution simply enumerated the details of these delegations; but again these were nothing more than delegations, which were subject to the final authority of the state's citizen-voters, who could overrule their state and federal government at any time; i.e. the state's citizen-voters, were the supreme power over their respective nation-state.