This actually is pretty simple to understand.
Imagine a population of 4 people, 2 men and 2 women. They pair off, and each pair has two kids, so that the next generation has 4 people as well. The first couple has 2 boys, the second couple has 2 girls. When the next generation pairs off and has kids, none of those kids will inherit the Y chromosome of the second male, because all his kids were girls. Every Y chromosome from now on will be inherited from the first male. And it doesn’t mean that every other male died, or didn’t have any kids, just that his kids were all girls.
Now extend that to thousands of people and thousands of generations. When a man fathers only girls, his Y chromosome doesn’t get passed on. Yet Y chromosomes have to pass on from generation to generation, unless one generation is so small that no boys are born, and the species becomes extinct. Eventually in any interbreeding population with multiple Y lineages, one Y lineage will go extinct if through random fluctuations every male who has that Y happens to have only daughters.
So, one guy 60,000 years ago fathered a boy who fathered a boy who fathered a boy who fathered a boy, and even though there were plenty of other men around who also fathered boys, at some point in the family of each of those men no boys were fathered, only girls, and their Y chromosome lineage became extinct.
If we look at the family tree of every human being on earth, they’ll have a branch for their father and mother, then their father’s father and mother and their mother’s father and mother, and so on. So everyone on earth has an ancestor who was an unbroken paternal ancestor, just like everyone on earth has an ancestor who was an unbroken maternal ancestor. You father had a father, his father had a father, his father had a father, his father had a father, and so on, back to the first sexual organisms. At the last time everyone’s father’s father’s father’s father’s father’s father was the same guy was 60,000 years ago, according to this study. So this is the reason that although Y lineages can go extinct, of course every Y lineage can’t go extinct.
And the reason we can tell this, is that Y chromosomes are passed on exclusively along the paternal line, just like mitochondria are passed on exclusively along the materal line.
And note that there were certainly other individuals who would be the paternal ancestor of all living humans, it’s just that this guy 60,000 years ago was just the latest one. Anyone earlier than him who was a paternal ancestor of all humans would have passed on their Y chromosome through him, and so it’s not possible to determine any other earlier common ancestors.
Of course, note the confounding factor that the Y chromosome that determines sex doesn’t always stay the same chromosome. For various complicated reasons, the chromosome that determines maleness loses more and more genes until the only things left are the genes that determine maleness. Then that tiny stub of a Y can get fused with another chromosome, and that chromosome becomes the new Y, and eventually all the genes get whittled away again. So the Y lineage doesn’t go all the way back to the first sexual organisms hundreds of millions of years ago. I’m not sure how long the human lineage has had our current sex determining chromosomes, but it hasn’t been that long.