Why is it true to say that “…the most lasting effects of the 1960s are to be found in the realm of 'the personal' rather than “the political?”
This is (arguably) true because the 1960s did not change politics in any fundamental way. They did not bring about the dominance of liberal ideas and they did not move the country towards a fundamentally different type of government.
Instead, the major changes from the '60s were personal. Personal life really has changed in many of the ways that were contemplated in the '60s. Women today are much more equal to men than they ever were before. The "sexual revolution" has in many ways taken hold and the idea that sex is only for procreation is more or less dead. The idea that people will only have sex within the setting of a marriage is also much, much less prevalent than it once was. Things are shown and said on TV that were unthinkable in the 1950s.
All of these sorts of changes are much more "personal" than they are "political." Our politics are not that strange and unfamiliar to people who were adults in the '50s and '60s. Our personal lives and our culture are much stranger and less familiar.