How did the Tariff of 1816 highlight the nation’s growing sectionalism?

Because the North approved it, while the South opposed it.
The tariff of 1816, like all tariffs, was a tax on imports. Before the income tax existed, this was a way for the U.S. government to raise revenue.
The southern economy was based on agriculture, chiefly cotton and tobacco. They really didn’t make much of anything, so they had to import almost everything they needed. They complained, with some justification, that they were therefore paying more than their fare share of the tariff.
In the North, the Industrial Revolution was really beginning to take off. Factories were being built and products produced. The North made things, which means they did not have to import as much.
Slavery is at the root of the problem. The South was becoming more and more dependent on it. And because slaveowners were making money, they didn’t want to change, so they didn’t want to industrialize. Very few factories would be built in the South.
The tariff was just another illustration of the problem of slavery.