How would the Second Amendment give checks and balances to the federal government?
There are at least two ways to interpret this question.
First, it could be asking how the Second Amendment creates checks and balances within the federal government. In this case, it does so by giving the judicial branch the power to check the actions of the legislature in some regards. For example, Congress passed a gun control law that affected Washington D.C. The Supreme Court struck that law down because it violated the Second Amendment. Thus the Second Amendment ended up causing the judicial branch to overrule the legislative branch. This is an example of checks and balances.
Second, the question could be asking how the Second Amendment gives the federal government power to check the states. Here, the answer is that the federal judiciary can check the actions of state legislatures by declaring laws they pass to be unconstitutional. For example, federal courts have struck down parts of a Chicago law (city governments are, in essence, part of state governments) that placed strict restrictions on who could own a hand gun in the city. By doing this, the federal government was using the Second Amendment to exert a check on a state government.