I shouldn't be surprised any more when running into this statement; it appears that it's actually taught in schools in the US or something. But it's still highly misleading, and gives a very confused picture of the concepts involved.
The statement only makes sense if you use Aristotle's original definitions for "democracy" and "republic." These referred to really existing polities in ancient Greece. Such polities, or even anything resembling them more than very superficially, no longer exist. Therefore, it's hardly surprising that the definitions we normally use -- both in regular conversation and in academic discourse -- have shifted a quite a lot.
In modern usage, "democracy" means a principle of government, and "republic" means a form of government. "Democracy" means a government that is "of, by, and for the people." "Republic" means a form of government that includes some kind of parliament and either a prime minister or president that are chosen through elections. The phrase for Aristotelian democracy -- the way it's used in your statement -- is "direct democracy," and it's more of a concept than a description of any really existing polity, even Switzerland which perhaps comes closest with its incessant plebiscites.
The two overlap, but by no means perfectly.
For example, Sweden is a democracy, because political power is vested in the parliament, which is chosen in free and fair elections, but it's not a republic, because the titular head of state is the king, which makes it a monarchy.
OTOH North Korea is a republic, but it's not a democracy: it has a prime minister and parliament, but these are chosen in elections that are anything but free and fair, which means that the electorate is effectively completely powerless.
In other words, the USA is a democratic republic, whereas the People's Republic of Korea is a non-democratic republic, and the Kingdom of Sweden is democratic without being a republic. Clear?