Kissinger is suggesting that Nixon's perception of being an "outsider" helped to drive he and his policies in order to find this appreciation. Nixon's background was outside of the establishment, being born a farmer's son in rural California. He was not born of wealth or privilege and went to local colleges in California, as opposed to the "established" universities of the nation. His "outsider" status was most evident in his presidential election challenges with then- Senator Jack Kennedy, born of wealthy background and a staple in the Harvard and East Coast community. Once Nixon ascended to political power, Kissinger's argument suggests that Nixon never lost sight of his "outsider" status and sought to be "embraced" by this established order of academics, wealthy, and those perceived to possess power. It is a critique that might make sense, but makes some assumptions. The first would be that Nixon's perception of "the establishment" never really changed, though so much of America did change at the time. During Nixon's ascendancy and reign of power, America changed massively through social dissent and through political legislation and action. It seems odd that despite all of this change, Nixon's supposed desire to be accepted by "the establishment" did not waver. Additionally, Kissinger's direct association with "the establishment" in his work at Harvard might also be a conflict of interest in making the claim. If Kissinger is suggesting that Nixon was victimized by his own desire to be accepted by "the establishment," then his selection of Kissinger, a member of the Harvard community, might be seen in a different light, in that Kissinger was not selected on his merits but rather for his affiliation. This is to say that while Kissinger is not wrong in his assertion, it is important to keep in mind that the complexity of the Nixon Presidency should not be reduced to solely base and conjecture- driven psychological perceptions.
The Eastern Establishment was composed of the elite of society and politics as well. They were the bipartisan elite of the East Coast, predominately WASP (White Anglo-Saxon Protestant); men such as the Rockefellers and the Henry Cabot Lodges were among this group. These were the men who created institutions and dominated the making of foreign policy from the Spanish American War through the Vietnam Conflict.
Richard Nixon, who was from the West Coast, was raised as a Quaker; his father was a service-station owner and grocer. He was constantly at war with the media which he felt was controlled by this Eastern Establishment. Of course, Nixon lost his battle with the media when Walter Cronkite, CBS news anchorman, resurrected the story of the burglarly of the Watergate hotel from the back pages of The Washington Post, known for being a liberal newspaper.
I think that when Kissinger says this he is referring to the "insiders" of American politics in those days -- the people who have power and money and come from families who had been rich for a long time.
Richard Nixon was very much not part of this "establishment." He came from a very modest background. This meant that he was not part of this elite who had gone to the prep schools of the East Coast and then to an Ivy League college.
Some people say that this need to prove himself drove Nixon to be as negative and angry as he often was.