Supporters of abortion rights won a huge and surprising victory on Tuesday in one of the most conservative states in the country, with Kansas voters resoundingly rejecting a constitutional amendment that would have let state legislators ban or significantly restrict abortion.

Results were still coming in as the night wore on, but with more than 90 percent of ballots counted, the pro-abortion-rights side was ahead by about 18 percentage points, a staggering margin in a state that voted for President Donald J. Trump in 2020 by a margin of just under 15 percentage points.

Here is a look at what happened.

Abortion opponents underperformed even in conservative areas.

Going into Election Day, many observers believed the outcome of the referendum would be determined in increasingly Democratic areas like the Kansas City suburbs — that is, by whether enough voters turned out there to compensate for the very conservative lean of the rest of the state. But abortion opponents did surprisingly poorly even in the reddest places.

Consider far western Kansas, a rural region along the Colorado border that votes overwhelmingly Republican. In Hamilton County, which voted 81 percent for Mr. Trump in 2020, less than 56 percent chose the anti-abortion position on Tuesday (with about 90 percent of the vote counted there). In Greeley County, which voted more than 85 percent for Mr. Trump, only about 60 percent chose the anti-abortion position.

We can talk about the cities all day long, but Kansas is known as a rural Republican state for a reason: Rural Republican areas cover enough of the state that they can, and almost always do, outvote the cities. The rejection of the amendment has as much to do with lukewarm support in the reddest counties as it does with strong opposition in the bluest ones.

Yes, the swing areas swung left.

Certainly, though, the cities and suburbs deserve some credit. The comparatively slim margins of victory for abortion opponents in western Kansas left the door wide open, but abortion rights supporters still had to walk through it, and they did.

Wyandotte County, home to Kansas City, Kan., voted 65 percent for Joseph R. Biden Jr. in 2020, but 74 percent for abortion rights on Tuesday. Neighboring Johnson County, the state’s most populous, voted 53 percent for Mr. Biden but more than 68 percent for abortion rights.

What was striking, in fact, was the degree to which the picture was similar everywhere. From the bluest counties to the reddest ones, abortion rights performed better than Mr. Biden, and opposition to abortion performed worse than Mr. Trump.

Turnout was high.

We won’t know exactly how many people voted, much less their partisan breakdown or demographic characteristics, until the results are fully counted. But we can already say that statewide turnout was much higher than expected — nearly as high as it was in the last midterm election.

Roughly 940,000 Kansans voted in the referendum, according to preliminary New York Times estimates, compared with about 1.05 million people in the November 2018 midterm election. The gap between turnout in primaries and general elections is usually much larger than that.

Before Tuesday, the Kansas secretary of state’s office predicted turnout of about 36 percent. But as voting ended, Secretary of State Scott Schwab told reporters that anecdotal evidence indicated turnout might hit 50 percent, an extraordinary increase over what was expected. The Times’s 940,000 estimate would mean 49 percent turnout.

The voters who would have been expected to show up on Tuesday, under normal circumstances, would mostly have been Republicans. That is not only because registered Republicans significantly outnumber registered Democrats in Kansas, but also because most of the contested races on the ballot were Republican primaries, giving Democrats little reason to vote — except to oppose the constitutional amendment.

The Supreme Court scrambled abortion opponents’ calculus.

Abortion opponents’ strategic decisions around the amendment started with the choice to put it on Tuesday’s ballot in the first place. The primary electorate was expected to be small and disproportionately Republican, and it seemed like a reasonable supposition that the amendment would have a better chance of passing in that environment than on a general election ballot.

The overturning of Roe v. Wade in June upended that strategy, turning what might otherwise have been an under-the-radar ballot measure into a nationally scrutinized referendum on abortion rights. Many voters might previously have seen the stakes as theoretical: If the U.S. Constitution protected abortion rights, how much did it really matter whether the Kansas Constitution did? But then the Supreme Court undid the first part of that equation, and Kansas abruptly became an island of abortion access in a sea of Southern and Plains states banning the procedure.

Groups on both sides blanketed the state with millions of dollars in advertising. Democrats who would otherwise have stayed home, knowing their party had few competitive primaries on the ballot, turned out specifically to vote against the amendment. Supporters of abortion rights were gripped with that great political motivator: anger.

On Tuesday, the results were clear.