More can always be said, but this older answer which focuses on 1876 election may be of interest, which I'll repost here with some slight updates. It doesn't look at 1896 specifically, but is contrasting the late 1800s with the early 1900s, and the impact of Jim Crow and other voter suppression, so is quite relevant.
To start things off, it is important to keep in mind that turnout was consistently high in the late 1800s. If you look at this chart you can see that while 1876 was the highpoint, it wasn't exactly an anomaly, and voter turnout was consistently high in the elections preceding it. It sticks out because, although the drop wasn't immediately afterwards, it certainly preceded the continuing decline in voter turnout that would begin precipitously around 1900.
So why did voting decline in that period? Well, one of the most simple reasons (but not only, of course) to look at is Jim Crow. While under Reconstruction, black men (women being generally deprived of the vote) could, for the most part, go to the polls and exercise their right to vote, this began to change after Reconstruction was ended (not to say it didn't happen before, just not as effectively), and the Redeemer governments worked through various means to disenfranchise vast swathes of voters in the American South. The effect of this can't be underrated. While in the 1876 election the South saw turnout roughly comparable to the rest of the country, at 75 percent, vote suppression methods such as literacy tests not to mention outright fraud, saw the turnout decline to 46 percent at the turn of the century. By the 1924 election, 19 percent of those theoretically eligible to vote were actually showing up at the polls. And to be sure, while the primary target was black voters, many poorer, illiterate whites were disenfranchised too, despite "loopholes" to grandfather many of them in. In Louisiana, for instance, while 90 percent of black voters were barred from the polls, 60 percent of whites were as well. While Jim Crow should absolutely be understood as primarily a racial regime, it was quite oligarchical as well, with power being concentrated in the hands mostly of upper-class whites, who wanted to share it with no one.
This allows us to circle back somewhat though to look at 1876, and why it would be slightly above the average of the time though. During the Reconstruction era there were real efforts to mobilize poor voters of both races by the Radical Republicans. The example I'm most familiar with was that led by Mahone in Virginia whose Readjuster movement controlled the state for a brief time in the late 1870s-early '80s, propelled by populist support from a coalition of black voters and poor whites. I won't spend to much time on him as I've written about him before here but the short of it is that in the post-war era, but before Jim Crow laws took hold, we can see a lot of political agitation that struck at the white Democratic establishment in the South that was attempting to reclaim power, and that for a time they enjoyed some successes. The 1876 election in particular we can look at as a watershed, with both sides of the argument over Reconstruction seeing heavy stakes. And of course Tilden won the popular vote, but lost anyways, as part of a deal that did end Reconstruction anyways. That cessation meant the evaporation of the Federal protections that allowed those insurgent political movements to compete on a roughly level playing field. Changes weren't immediate, and varied state by state - in Virginia for instance the Readjusters remained in power until 1883, when race riots days before the election were used by the Democrats to stoke voter fears - but it nevertheless meant that the suppression of the black vote and the poor white vote was able to start, a process which wasn't immediate, and took time to take full effect.
So having looked a little at the election of 1876, and how it is a little higher than normal, but not too out of line for the period, this leads to another question... Why was 1872 a very noticeable anomaly at 71.3%, bounded in 1868 by a turnout of 78.1%, and of course 1876 at 81.8%?
Let's start in 1868. That year, Grant ran against Horatio Seymour, a northern Democrat from New York, and won with a fairly respectable vote lead. He was especially buoyed by a strong performance in the South, winning many of the former Confederate states. How though? Well, an astute observer might notice that despite a much higher turnout percentage, the actual numbers of voters was lower, which would be somewhat counter-intuitive. But, while I have been unable to find any sources which explain specifically how that number is calculated, if it is counting eligible voters against turnout, it seems likely that it is accepting the disenfranchisement of many white former rebels in the Southern states as legal and proper, and not counting them in the rolls of eligible voters. Certainly, it was the lack of their votes which, unable to balance out the newly enfranchised black vote in the South, helped to support his campaign down there.
This stands in comparison to the 1872 election, where those former rebels had since been reenfranchised, and thus counted in the rolls of eligible voters. But many voters nevertheless chose to stay home, why? Well, check out who was running! Grant was seeking reelection, but his opponent was Horace Greeley - a Republican, not a Democrat! He was riding the Liberal Republican ticket, a splinter party from the Republicans, dissatisfied with Grant and much of what he had done in his first term. They lacked an entirely cohesive platform but generally stood, at the least, for playing nice with the South, and ending or softening the various Federal policies directed there. Greeley had been one of the most forceful voices in the Republican party for sectional reconciliation, and would win its nomination, but it wasn't straight forward, and caused a good deal of acrimony. At the Party Convention which nominated him, he had gone into is not as the favorite, beating out Charles Francis Adams after six ballots. There was enough bitter disagreements after his victory speech and declaration of platform that Adams supporters mostly just went back to the Grant camp, even if begrudgingly, unwilling to support the "turncoat and traitor", representative of the lack of cohesion in the movement from the start.
As for the Democrats... they simply didn't run a candidate. As newspaper editor Theodore Tilton wrote in aptly summing up the sentiment followed:
Since the Democratic party pledges itself to abide by the constitutional amendments [...] and since it wants universal amnesty [...] why not therefore let the better class of Democrats unite with the anti-Grant Republicans?
But the party was dominated by the Northern Democrats, and perhaps the platform was one they could support, but not necessarily one that made voters in the deep south happy, or at least enough so to nominally vote for a non-Democrat. In the end, it was a fairly tepid election, between two candidates who didn't quite endear themselves well. Grant waltzed to reelection with a 11.80 percent margin on the back of an electorate that perhaps wasn't all in for him, but couldn't get behind Greeley's "motley" mix of appeals. What it did represent though was the growing dissatisfaction within some parts of the Republican party, who tired of Reconstruction which they were becoming convinced was a failure. It was, of course, a conflict that would underpin the next election, as already noted, but with an actual context between Republicans and Democrats, rather than divisions within the Republican Party alone.
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Excellent reads. It’s amazing how much of this is glossed over in school and even college level courses in the US. This is must know information .
If you want to read in detail about it, Keyssar's The Right to Vote: The Contested History of Democracy in the United States is a good overview, if overly descriptive at times. His overarching argument is that although the United States was unique in gaining a very extensive franchise from a very early period - something like 70-80% of adult white men able to vote by 1800 - this was achieved before industrialisation and the threat of an organised working class. Once that occurred, with the social changes accompanying it, there was a general pushback against democracy - "the terms of public discussion were being set by men who believed that universal suffrage had failed, that it was neither viable nor desirable in the socially heterogeneous, industrial world of the late nineteenth century" - this being most prominent in the period after Reconstruction.
The most egregious example of this is the disenfranchisement of African Americans in the South with Jim Crow laws. In 1890 Mississippi, less than 9000 out of 147,000 voting age blacks able to vote, or Louisiana went from 130,000 registered blacks to 1342 from 1896-1904. But this wasn't a purely racial picture, but a class-based one. The sort of restrictions that disenfranchise black voters are likely to target poor whites too, and helps explain why Mississippi went from 70% turnout in the 1870s to only 15% for most of the early 20th century. And the United States enjoyed a surge in immigration at this time, and the presence of ~15% of the population born in another country, with another significant percentage being the children of immigrants, and often not speaking English or looking much like 'old-stock' Americans, solidified reaction to them.
Furthermore, as Keyssar draws attention to “depending on the state or city in which he lived, a man could be kept from the polls because he was an alien, a pauper, a lumberman, an anarchist, did not pay taxes or own property, could not read or write, had moved from one state to another in the past year, had recently moved from one neighborhood to another, did not possess his naturalization papers, was unable to register on the third or fourth Tuesday before an election, could not prove that he had canceled a prior registration, been convicted of a felony, or been born in China or on an Indian reservation”.
We tend to think of these kind of restrictions as relatively minor, but when analysed together, it had a significant effect on turnout, and was clearly class(and immigrant)-discriminatory. A few examples demonstrate this point. In 1921 New York – the city with the largest immigrant population in the nation – followed many other states in enacting a constitutional amendment to enact a literacy requirement. Given 25% of men who took army literacy test during WWI failed it, the effects of such a policy were enormous. Residence and registration requirements operated in a similar way. The number of people who lived in Boston at any point in 1880s was 3x greater than its highest population at a single time (fairly typical for many cities at this time), the vast majority of this mobile population consisting of members of the working class who would now find it much harder to vote. Another example is the introduction of restrictive voting policies for immigrants. There is nothing in the constitution saying that aliens can't vote, and all restrictions on it comes from the states. The picture varied over the 19th century (generally frontier states would have permissive alien voting laws to encourage settlement, and in the East it was more restricted) but by the early 20th century the tide had turned decisively for restrictions - Arkansas being the last state to allow it, changing that by 1926.
In total, most of these new restrictions were only introduced in the late 19th and early 20th century, and they were disproportionately targeting immigrants and the working class. This goes a significant way to explaining how turnout dropped so much by the 1920s.