According to James Madison, why does America have a republic instead of pure democracy?

In his essay "Federalist No. 10," James Madison lays out his feelings about direct democracies and republican governments. In this essay, he describes his arguments for the benefits of a republic—which the United States has—over that of a pure democracy and defends the notion as described in the Constitution, which was written only two months earlier. Madison contends that the country was built as a republic because a pure democracy would lead to too many problems in a nation the size of the United States.
First off, Madison writes that a pure democracy, where the people rule directly, would have "no cure for the mischiefs of faction." In other words, people would make decisions based too much on self-interest, or the interests of their specific group, and not for the greater good of the country as a whole. Specifically, Madison was concerned that the unpropertied class might work to take and redistribute the property of the propertied class if left unchecked by representatives. More generally, Madison argues, by electing representatives, our leaders will hopefully make decisions based on the wisdom and love of country for which they were elected, something that the general populace might lack.
Madison continues to argue the virtue of a representative government in a large republic. Unlike a small democracy, in a large republic such as the United States, representatives can

extend [their] sphere, and you take in a greater variety of parties and interests; you make it less probable that a majority of the whole will have a common motive to invade the rights of other citizens; or if such a common motive exists, it will be more difficult for all who feel it to discover their own strength, and to act in unison with each other.

So it is clear from his writing that James Madison supported the representative republic as laid out in the Constitution. He thought the form of government to be more sustainable over the long run and suitable for a country with the size and composition of the young United States.
https://avalon.law.yale.edu/18th_century/fed10.asp