The old, time honored, adage applies here: It's the economy stupid. The US economy is the largest in the world and by some distance too, at about roughly the same size as the next three largest economies combined(China, Japan and Germany, in that order). We live in a world where economic heft is the single most important determinant of your position in the global food chain and the US is light years ahead of any potential competitor. Secondly, to address the 'free world' point, an understanding of the US' role in shaping and maintaining the post WW2 international order is necessary.
The term free world is a thinly veiled reference to the west and a handful of US allies in the Asia-Pacific region which, combined, constitute a huge chunk of the global economy entirely disproportionate to their share of global population, combined with this bloc's collective ability to shape and influence events across the globe. Most of these countries are liberal democracies characterized by rule of law, personal freedom of the individual and generally egalitarian societies with minimal interference from the state in the lives and economic freedom of their subjects. The term was first popularized during the cold war to contrast the US and it's allies from the other competing bloc led by the Soviet Union which stood for authoritarian governments, suppression of human rights and minimal, if any, personal and economic freedom.
Now that the free world bit is out of the way, lets consider why the US has more of a claim to its leadership than any other contender. The single most important reason for that may be that the other members of the liberal, western dominated world order themselves defer to the US when it comes to taking charge and leading the bloc as a whole, be it militarily or on international fora where the rules by which the world operates are decided(the likes of WTO, various UN committees etc).
The US has most of these privileges not because it's inherently better suited to lead the world but because their economic and military might overshadows all others so comprehensively that they simply have no competitors and their network of alliances mean that almost everyone who's anyone is part of their sphere of influence. On that note, it bears mentioning that alliances matter a lot. The EU and US, close allies, account for almost half of the global economy despite being home to about 10-11% of the world's population. The importance of alliances is evident by the fact that China, despite being an economic behemoth in its own right, has disproportionately little influence globally because it's basically them and a handful of tier 2 and tier 3 countries like Russia ,which still maintains a certain level of independence, Pakistan and North Korea.
As for why the leaders of Pakistan or India can't claim that title, can you imagine the PM of Pakistan, a man who isn't even the leader of his own country (that would be the army chief and, to a lesser extent, the DG ISI), trying to impose economic sanctions on, say, India and demanding that the rest of the world follow suit or else, like the US did with Iran? The world may plunge into a leadership crisis as several major world leaders run the risk of laughing themselves to death in such a scenario. Ditto for India, a country that doesn't even have enough weight to throw around to impose it's will on a borderline banana republic like Pakistan much less project hard and soft power simultaneously on six continents and influence the way events unfold. The second issue with India is that while their overall economy is large enough to classify them as a major power, they're still a third world country and they act like it on the international stage.
