He didn't, in the way that most people use that term. He proposed multiple plans to expand the size of the Supreme Court beyond 9,but they were defeated. What he did was successfully put up a large number of justices, which makes sense since he was a 3.5 term president.

FDR had just come out of the massively successful 1936 election; the Supreme Court had begun to strike down various aspects of the New Deal as unconstitutional, such as the Agricultural Adjustment Association (AAA) and National Recovery Administration (NRA).

The Supreme Court is the one unelected branch of the government: they serve until they step down or die. Roosevelt, feeling impatient and confident because of the massive margin by which he won the election, proposed that Supreme Court justices step down at the age of 70 and retire. As [History describes,] (http://www.history.com/this-day-in-history/roosevelt-announces-court-packing-plan) "If a justice refused to retire, an 'assistant' with full voting rights was to be appointed, thus ensuring Roosevelt a liberal majority." This would have resulted in as many as 15 judges at the time, as History further reports, but could max out at 18, if each judge were to be over 70.

Some of FDR's justifications, such as that he wanted to lighten the workload on the Court, were demonstrably false. If memory serves, one member of the Supreme Court, which disagreed with the plan, argued, with evidence, to show that they were not overworked, from an objective viewer's perspective.

FDR was wrong. Even though he'd won the election by a massive margin, the public did not react positively to government overreach and tampering with the Supreme Court, resulting in what some consider to be the most humiliating defeat of FDR's presidential career. The bill he proposed was overwhelmingly voted down.

Sources:

Conrad Black, Franklin D. Roosevelt

Mitch Hiltzik, The New Deal: A Modern History

H.W. Brands, Franklin D. Roosevelt