tl;dr
The opening sentence succinctly summarizes the historical distance between Edwardian and postwar Britain, as well as the psychological distance between Leo's schoolboy and adult selves.
Deets
The 1950s world of the narrative present of The Go-Between is far removed from the turn of the century world of Leo Colston's recollections. Socially and geopolitically, Britain after the two wars was a very different country from Edwardian Britain. The loss of the "automatic deference" given to the upper classes in the social sphere that @Kate Bunting notes in her answer was paralleled by Britain's diminished status in the geopolitical realm. Britain was effectively bankrupt and rapidly losing its overseas colonies. The United States had taken over from the British Empire as the pre?minent global power. When Leo says "The past is a foreign country," the statement is practically literally true when comparing pre- and postwar Britain. The schoolboy Leo's own anticipation of the coming century is heartbreaking in retrospect:
The year 1900 had an almost mystical appeal for me; I could hardly wait for it: "Nineteen hundred, nineteen hundred," I would chant to myself in rapture, and as the old century drew to its close, I began to wonder whether I should live to see its successor. I had an excuse for this: I had been ill and was acquainted with the idea of death; but much more it was the fear of missing something infinitely precious—the dawn of a Golden Age. For that was what I believed the coming century would be: a realization, on the part of the whole world, of the hopes I was entertaining for myself.
Hartley, L. P. The Go-Between. 1953. New York: Stein and Day, 1980. p. 6.
But as this passage suggests, the schism between the past and the present is more than historical. Leo's na?veté about the coming century encompasses more than the cataclysmic changes that will occur in the world. His optimism about his own fate too is misplaced. The opening sentence highlights the psychological disconnect between the narrator and his schoolboy self. The events of Leo's stay at Brandham Hall leave him traumatized to the point of losing his memory and the faculty of speech (pp. 305–306). He recovers those quickly enough, but he never comes to terms with the trauma itself. The adult Leo says of "the events of those nineteen days in July":
I had kept them buried all these years, but they were there, I knew, the more complete, the more unforgotten, for being carefully embalmed. Never, never had they seen the light of day; the slightest stirring had been stifled with a scattering of earth.
My secret—the explanation of me—lay there. I take myself much too seriously, of course. What does it matter to anyone what I was like, then or now? But every man is important to himself at one time or another; my problem had been to reduce the importance, and spread it out as thinly as I could over half a century. Thanks to my interment policy, I had come to terms with life, I had made a working—working was the word—arrangement with it, on the one condition there should be no exhumation. Was it true, what I sometimes told myself, that my best energies had been given to the undertaker's art? If it was, what does it matter? Should I have acquitted myself better, with the knowledge I had now? I doubted it; knowledge may be power, but it is not resilience, or resourcefulness, or adaptability to life, still less is it the instinctive sympathy with human nature; and those were qualities I possessed in 1900 in far greater measure than I possess them in 1952.
ibid., pp. 16–17
The 1950s Leo is almost a ghost, his real self buried under his fervid repression of the events at Brandham. The human qualities his younger self had are lost to his older self. In an imagined dialogue between his present and schoolboy selves, the adult Leo blames the child for this loss:
"This cindery creature is what you made me."
To which he might reply: "But you have had half a century to get over it! Half a century, half the twentieth century, that glorious epoch, that golden age that I bequeathed to you!"
"Has the twentieth century," I should ask, "done so much better than I have? When you leave this room, which I admit is dull and cheerless, and take the last bus to your home in the past, if you haven't missed it—ask yourself whether you found everything so radiant as you imagined it. Ask yourself whether it has fulfilled your hopes. You were vanquished, Colston, you were vanquished, and so was your century, your precious century that you hoped so much of."
ibid., pp. 18
The explicit connection Leo draws between the way the century itself has unfolded and the way the events at Brandham Hall have shaped his personality is implicit in the novel's opening sentence. The dialogue between the two dramatizes the foreignness of Leo's past to his present: the schoolboy Leo "did things differently" from the adult Leo indeed.