The Voting Rights Act of 1965 relates to federalism because it puts the federal government in charge of the voting procedures in a number of states and counties, the bulk of which are in the South. This is one reason why the act is being challenged in court.
In our system of federalism, some powers are given to the national government while others are kept for the states. Most of the time, the procedures by which elections are run is something that is left to the states to decide. For example, some states have elections in which all voters vote by mail while other states allow mail-in ballots in only very limited circumstances.
In the 1960s, it was found that many states had been using voting rules to keep African Americans from voting. The Voting Rights Act of 1965 tried to stop this by taking the power to set up voting arrangements away from those states and giving it to the federal government. This is why this law relates to federalism; it takes a power away from the state and gives it to the federal government.