Why was it significant that Thomas Jefferson became president and the Anti-federalists came to power?
First, please note that by 1800, Jefferson's party was called the Republicans or the Democratic-Republicans and was not really referred to as the Anti-federalists. That said, the major significance of this change was that it was the first time that power was handed over from one party to another in the United States. The fact that it was handed over peacefully was hugely significant.
In many new countries, whoever takes power at independence holds on to power with a death grip. This was not the case in the US either personally (George Washington stepped down voluntarily) or in terms of parties (the Federalists did not try to prevent Jefferson from becoming president once he had won the election. Because the Republicans won and the Federalists gave way peacefully, the US was set on the path to true democracy and a stable government.
Jefferson's win also represented the start of a move towards more democracy in the US and way from Federalist elite rule. This would not be complete for another two decades after Jefferson's win, but his win was a definite step in that direction.