There are three main factors here. (1) Eisenhower himself and his civil rights beliefs. (2) The nature of the Democratic Party in the 1950s and its complex makeup. And (3) the Cold War, and what it meant for domestic policy.

Background: the Democratic and Republican Parties at mid century are divided more along lines of the government's role in the economy, the size of the federal government, and differing opinions about foreign policy. Race is certainly a powerful animating force, but it is not the issue that divides the parties. The GOP had long abandoned the banner of black rights (which Pres Taft stated nigh explicitly in his inaugural address), but the Democratic Party had not wholeheartedly accepted it. This is because it was essentially divided between Southern dixiecrats and Northern progressives, who had generally fallen behind FDR in a quest to rehabilitate the American economy. But the FDR years only papered over the very real differences in the party regarding civil rights, hence why many New Deal programs had notable exemptions for issues that would have threatened Southern racial hierarchies. These differences made real progress on civil rights extremely difficult with FDR and the Truman administrations, despite the fact that both Democratic presidents were personally sympathetic to the cause of black rights. There were simply too many Southerners in the party who objected. Southern dixiecrats were also becoming increasingly independent, aligning with the GOP to shut down key legislation in Congress, which made it harder for the Democratic Party to shepherd civil rights bills through Congress.

The lack of progress disappointed many Black activists and civil rights leaders, who saw immense hypocrisy between the soaring rhetoric of the presidents and their actual difficulties enacting meaningful changes in reality.

Eisenhower himself was a far cry from what we in the 21st century might expect from a GOP president. David Nichols argues that Eisenhower was relatively progressive on civil rights issues. His record included supporting the desegregation of the military, of DC, and appointing justices and judges who were pro civil rights. The Brown v Board of Education ruling happened during his administration, to which Eisenhower was happy to enforce, with the infamous deployment of paratroopers to a Little Rock high school to support desegregation as a notable example.

Eisenhower was not a starry-eyed dreamer--there were very real political considerations behind his pro civil rights alignment. The Cold War had put the US in a position where its soaring ideogy was very much in conflict with its domestic realities. The hypocrisy between liberal ideology and segregation was very much on display for the world to see, which weakened America's global position. Thomas Bortelsmann and Mary Dudziak have argued persuasively that American officials were aware of this throughout the Truman and Eisenhower administrations, and internally lobbied for substantive federal action to address minority rights as a matter of Cold War politics. Eisenhower himself also saw the electoral benefit of trying to rebuild old GOP ties to black voters.

Thus, in the 1956 election, Eisenhower won a significant portion of black votes because of his civil rights record, the internal coalition that stymied substantive change in the Democratic Party, and the pressures of a Cold War world. With the 1960s, however, America would soon see differing alignments in the political parties and who supported them.

Sources: David Nichols, A Matter of Justice. Taft's Inaugural Address (1909) Ira Katznelson, Fear Itself Thomas Bortelsman, The Cold War and the Color Line Mary Dudziak, Cold War Civil Rights.