Clinton came into office. At the start of his first term, he presented his plan to "balance the budget", which was to make small cuts for the next eight years, then large cuts thereafter - after his second term ended!
Then came 1994, and the "Contract With America". Republicans took over Congress. Their next budget made significant cuts then, not after Clinton was out of office. That's how "Clinton balanced the budget" - by a Republican Congress forcing him to live up to his words.
Lest you think this is a partisan statement, let me note that Republican Congresses never balance the budget when a Republican is President. But the historical record is that it wasn't just Clinton's achievement - a Republican Congress had to make him do what he wanted to make the next president do.
That was an annual joke back then. Every year they'd present a budget that had spending increases in the first couple years, "balanced" by cuts in the "out years" 5-10 years in the future. Of course, the next year there would be a new budget that invalidated all the future cuts but kept the increases that had already taken effect.
Then at some point they just stopped pretending to balance it at all, and no one seemed to care much.
Then came 1994, and the "Contract With America". Republicans took over Congress. Their next budget made significant cuts then, not after Clinton was out of office. That's how "Clinton balanced the budget" - by a Republican Congress forcing him to live up to his words.
Lest you think this is a partisan statement, let me note that Republican Congresses never balance the budget when a Republican is President. But the historical record is that it wasn't just Clinton's achievement - a Republican Congress had to make him do what he wanted to make the next president do.