He did it by going to war. As left-progressive writer Randolph Bourne famously said at the time: "War is the health of the state."
Wilson initially did not want US involvement in WWI. Even after the sinking of the Lusitania in 1915, Wilson accepted German assurances that it wouldn't happen again. He didn't ask Congress for a delaration of war until April 1917, almost 3 years after the start of the war.
But once he entered the war, he went full force. His overall strategy had 4 main prongs, all of which remain significant 100 years later.
1)
Propaganda. Wilson created a "Committee on Public Information" headed by a former newsman named George Creel. Creel recruited 75,000 'four minute men' who spoke at theater intermissions and the like. The alliance between media and state remains strong today, with former political operatives like Tim Russert, George Stephanopoulos, and Bill Moyers moving to "journalism" jobs.
2)
War Socialism. This was a concept invented by Gen. Erich Ludendorff of Germany. The idea was to harness industrial might in service of war. The concept was copied by Lenin, and by Wilson's Treas. Sec'y Wm. McAdoo, who created the "War Finance Corporation." The newly enacted (1913) federal income tax zoomed from a top rate of 7% to 77% during the war. This was the start of the 'military-industrial complex' that Ike later warned about, and of course remains incredibly powerful in 2018.
3)
Curtailment of liberties. Wilson put numerous people in jail merely for speaking against the war. The most famous was the Schenk case, which went to the Supreme Court, and whose conviction for distributing anti-war pamphlets was upheld. Wilson signed into law the Espionage Act of 1917, which was used extensively by the
Obama admin to harass reporters and leakers. There have been suggestions that Pres. Trump will continue the trend. Curtailment of liberties generally was a long running feature of post-Wilson America. Of the 10 amendments comprising the Bill of Rights, only the Second Amendment was actually strengthened in the past hundred years, thanks mostly to the NRA. The First Amendment remains under constant attack, and the Fourth Amendment is essentially null and void.
4)
US dominance over world affairs. Wilson went into WWI full force. The US army went from 200,000 troops at the beginning of 1917 to over 4 million at the end of 1917. Since the US was primarily responsible for the war's conclusion, Wilson and the US took the lead in defining the peace. One hundred years later, this is more the case than ever. We all know the stats: US spends 3.5% of GDP on military; the median for NATO countries is 1.3% (despite their agreement that the number is supposed to be 2%). We spend more on the military than the next 8 countries combined. And so on.
These were four big trends initiated by Woodrow Wilson that remain important features of the American state 101 years later.