The principle of federalism resolved the issue of states versus national rights by establishing what rights belonged to states and what rights belonged to the federal government.
When the new US Constitution was developed, the founders considered the flaws of the Articles of Confederation. The Articles, which contained an extremely weak federal government, needed to be changed in order for the country to run effectively. This is why the federalist principles were implemented.
The federal government has the power to make national laws, appoint federal judges, declare war, make treaties with other countries etc. Any right not specifically given to the federal government belongs to the state governments. This is included in the Bill of Rights and ensures that there is a sort of balance between state power and federal powers.