Overall, a presidential inauguration is paid for through an amalgamation of private and public funds.
"In bureaucratic terms, the costs are shared by the Joint Congressional Committee on Inaugural Ceremonies, the Presidential Inaugural Committee, the federal government, and state and local governments," The New York Times explained in 2017.
The private money, footed by the Presidential Inaugural Committee, pays for the events — balls, concerts, dinners, etc. The bulk of Trump's inaugural committee money came from wealthy donors and corporations like Pfizer, Bank of America, and AT&T. The committee raised $107 million between Election Day and the inauguration. (Federal prosecutors in Brooklyn and Manhattan are reportedly looking into whether there were illegal donations made from foreign persons.)
The Joint Congressional Committee pays for and organizes the swearing in and the luncheon that follows; $1.25 million was appropriated by Congress for these events.
Security for the event is the biggest expense (could be more than $100 million), and the federal government pays for that.
Source: The New York Times