If their was enough of an objection to African Americans voting that Jim Crow laws were implemented to circumvent the 15th amendment, then how were its proponents able to get support from enough states to get the amendment passed?
So this is a great question that gets to the heart of political life post-Civil War, or, as it's commonly referred, the Reconstruction Era. There is some dispute as to when Reconstruction officially began, with some pointing to Lincoln's Emancipation Proclamation (1863) and its subsequent enforcement in captured confederate territory, while others argue it doesn't truly begin until the cessation of open hostility in 1865. Regardless of this dispute, it is widely agreed Reconstruction was dealt its final death blow in the Compromise of 1877 and the ascension of Rutherford B. Hayes to the White House.
But before we talk about the end of Reconstruction, we should talk about Reconstruction itself. The question of what to do with the Southern rebels after wars end was a headache that plagued leaders in Washington pretty much from the very beginning of the conflict, with what felt like about as many proposals floating around as there were members of congress — perhaps even more. For our purposes we can really put them into three column: redemptionist plans that sought rapid reintegration, radical plans that sought to fully reform the South on the backs of a newly freed black electorate, and moderate plans that sought a middle ground. As you can imagine, each philosophies subscribers hated the other with a passion.
The fundamental dispute hinged on what role, if any, black Americans would have in the new South. Democrats and some conservative Republicans advocated for a slow, piecemeal process of freedom and social integration. Radical Republicans, on the other hand, demanded a rapid process — immediate emancipation, suffrage, and economic support most frequently taking form in land redistribution. A keen student here might be able to suss out the moderate stance at this point, but for those unsure: they sought a fun little middle road between the two.
The first defining act of Reconstruction was the passage of the 13th Amendment, which was accomplished early in 1865 during a lameduck (the period of time between an election and the next class of representatives taking their seats) session of the House of Representatives, and is of course defined by this divide. A coalition of the Republican Party and a handful of Democratic defectors managed to pass the amendment, primarily on the grounds that all it did was free the slaves.
Following this, the end of the war, and Lincoln's assassination later that year, Andrew Johnson took office. Much has been said about what Lincoln's likely feelings on Reconstruction would have been, but we don't have to wonder all that much about Johnson — a Tennessee Democrat, he perhaps naively felt the nation was ready for rapid reintegration, and sought to restore the Southern States quickly. For him the question of civil rights was second to a return to normalcy, a feeling the (Radical) Republican congress did not share. The consequences of Johnson's belief in a rapid reconstruction can be seen most clearly in the creation of Black Code laws in Southern states, passed by Democrat dominated reformed Southern governments to police the conduct and behaviors of their black population. While laws like this were quickly banned by Congress, and policed by the formation of the Freedmen's Bureau, early Southern governments demonstrated they were all too willing to resist black freedom as much as they could. Johnson, for his part, largely ignored the Black Code and by most accounts encouraged Southern States to reject ratification of the 14th Amendment, which would extend citizenship to freedmen (including the vote).
In response to Johnson's perceived indifference to Southern recalcitrance, Congress stepped in. Dominated in both houses by a Radical Republican majority incensed at the murder of Lincoln and an utterly unrepentant South, they began a process of wresting control of Reconstruction from Johnson. They established a commission to oversee the process in 1865, and in a series of ever ratcheting disputes with Johnson ultimately impeached him, and passed the Reconstruction Acts of 1867. Perhaps most significant of these was the first, which suspended civilian government in all but one of the former confederate states, dividing them into six districts managed by the Union Army.
One of the primary objectives of this process was the protection of the newly freed black population, and guarantee their ability to vote in free and fair elections — their right as citizens of the United States, granted to them by the 14th Amendment. Under military administation new civilian governments were formed, new constitutions drafted, and one-by-one those once rebellious states were readmitted to the union. Initially this readmission was contingent on ratification of the 14th Amendment, and later the 15th Amendment.
SO, all of that to get to the point of this post: how was the 15th Amendment ratified? First it's important to note that the 15th Amendment did not grant the right to vote to freedmen, it protects their right to vote. That right itself was granted by the 14th Amendment, ratified a full two years earlier. This distinction was noted even then, and critics argued it was redundant as the 14th Amendment, they argued, already did that. With hindsight we know that was far from the case, as Radical Republicans suspected at the time. So by the time the 15th came up Southern Governments had undergone radical changes under the guidance of their military governors, and a newly elected coalition of white & black Republican office holders had taken power in these states, alongside various state-specific parties formed by working class whites that found common cause with the new black electorate. That coupled with the election of Ulysses S. Grant, changed the landscape. He prioritized the end of Military Reconstruction, and working with Congress secured the ratification of the 15th Amendment by attaching its ratification to readmission into the Union. The final holdouts of Military Reconstruction, Virginia, Mississippi, and Texas enshrined the 15th Amendment into their constitution as part of their readmission — and with that the 15th was the law of the land.
But what happened? Why did everything fall apart? As you might suspect, not all of this was popular — least of all in the South. Resistance to Reconstruction was sharp, perhaps most famously seen in the guerrilla violence of Nathaniel Bedford Forest and the Klu Klux Klan. "Defenders of white honor," the Klan and other groups like them tried their damnedest to reignite the flames of war, embarking on a campaign of terror targeting freedmen, Republican leaders, and the Union Army. While the Klan ultimately lost thanks to President Grants pretty stalwart commitment to the protection of Freedmen, they represent the first signs of fracture.
The political coalitions holding these Reconstruction governments were weak, often held together at the seems by Federal support from Grant, the newly formed Justice Department, and the Union Army. Grant for his part, while a strong defender of civil rights, was a weak president eternally embroiled in cabinet scandals. The Republican coalition slowly fell apart, popular support waining in the wake of Grants scandals. While Grant secured reelection in 1872, the damage was already done - seeing opportunity Southern Whites struck throughout the late 60's and early 70's. Black office holders were removed from office, some by Democratic political maneuvers, others via violent race riots. As part of this proces, Southern Democrats embarked on an influence campaign to sway Northern whites to their cause. They accused Radical Republican governments in the south of corruption, claimed black politicians lacked the ability to serve, and claimed they were simply doing their best to preserve public order by seizing power. Northern Whites, exhausted by what they felt was a never ending reconstruction, lost interest. With an economic panic in 1873, their sights were no longer on the South.
Emboldened by this indifference violence only continued to grow, with Southern Democrats gradually taking control of their states back from the Radical Republicans they labeled scalawags and carpetbaggers. The era of Radical Republican governance in the South officially came to an end with the elections of 1874 and 1876. In 1874 Democrat regained control of the US House for the first time since the start of the Civil War, and then in 1876 used their new power to negotiate a complete withdrawal of Union troops from their remaining Southern posts in exchange for awarding 20 in-dispute electors to Republican Rutherford B. Hayes in that years presidential election — rejecting their own party's candidate Samuel J. Tilden, who had won the popular vote. Having negotiated an end to the military presence, and taking control of Congress in the elections of 1874 and 76, Democrat Redeemers ushered in the Jim Crow era as we know it.
So with all of that said, the short version is this: the 15th Amendment passed thanks to the work of Radical Republicans in Congress imposing military reconstruction on the South, a president that prioritized its passage in the form of Grant, and a Southern political landscape briefly controlled by a coalition of black and white Republicans. The rise of Jim Crow in the South can be attributed to political fault lines that had been present throughout Reconstruction, fault lines that were exacerbated by a series of political scandals and economic crises, and ultimately secured through a campaign of violence and political intimidation that went ignored by a North preoccupied with other matters.
If you'd like to read more on the subject, I cannot recommend Eric Foner's seminal work on Reconstruction enough, Reconstruction: America's Unfinished Revolution, 1863-1877