Throwing so the answer is good
Throw an eye of ender and, while it hangs in the air before it drops, turn so your crosshair sits right on it. Note your position and angle (below), and pick the eye up if it did not break. Then walk a good distance sideways, at right angles to the way it flew, and throw again. The lines need to cross at a clear angle: two throws a few blocks apart point the same way and say almost nothing about distance. For a stronghold 1,500 blocks off, 300 to 500 blocks sideways is a good start. A third throw from somewhere else tests the first two. Each eye has a 1 in 5 chance to break, so bring spares.
Reading the angle from F3
Press F3 (Fn+F3 on some laptops). The line Facing reads like north (Towards negative Z) (-163.4 / -12.0): the first number in the brackets is the angle to use, here −163.4, and the second is how far up or down you look, which does not matter. Your position is the XYZ line; use X and Z. F3+C copies a line with both at two decimals, which is more exact; paste it into the box above. Angles here are the game's: 0 looks south, 90 west, 180 north, −90 east.
How the answer is worked out
Each throw is a line on the map from where you stood, in the direction you faced. With two throws the answer is where the two lines cross. With three or more they will not all meet at one point, so the page finds the point that fits them best: each line's miss is measured as an angle, since a small error in angle puts the line further off the further it reaches, and the total of the squared misses is made as small as possible. The ± figure is about how far off the answer may be, given how exact you say your angles are (or how much your throws disagree, if that is more); it gets smaller the more the lines cross at a right angle. The page also says how far each throw's line misses the answer.
Going there through the Nether
The Nether coordinates are the Overworld ones divided by 8 and rounded down, as for portals. Travel there in the Nether, build a portal, and throw again on the Overworld side: close up, a single throw shows the way, and an eye thrown within a few blocks of the stronghold drops down toward it rather than flying on. The eyes lead to where the stronghold starts, not to the end portal room, which you find by following its corridors.
The stronghold rings
Java Edition places 128 strongholds in 8 rings around 0, 0: 3 in the first ring between 1,280 and 2,816 blocks out, then 6 between 4,352 and 5,888, 10 between 7,424 and 8,960, 15, 21, 28, 36, and 9 in the last ring between 22,784 and 24,320. They are spread evenly round each ring. The diagram draws the rings, and if an answer falls between two rings, allowing for the ±, the page says so: a throw is probably off. Checked against Java Edition 26.3.
Bedrock Edition
Eyes of ender lead to the nearest stronghold in Bedrock Edition too, so the crossing lines work the same, but Bedrock does not show the facing angle, and it places strongholds differently (many sit under villages), so Java's rings do not apply there. The ring check is for Java Edition only.
Questions
Why is the ± so big?
Because the lines cross at a narrow angle. If the two throws were made close together, a tenth of a degree moves the crossing point a long way along the lines. Walk further sideways and throw again, or add a third throw from somewhere else; the ± shrinks as the lines cross more squarely.
It says my answer is behind me. What went wrong?
The lines cross on the side opposite where the eyes flew, which happens when an angle was typed with the wrong sign, or two throws were swapped. Check each angle against F3; a minus sign is easy to miss.
Do I use the angle of the eye or the angle I threw at?
The angle the eye flies, which is the one you face once your crosshair is on the eye as it hangs in the air. The direction you threw it in does not matter: the eye turns toward the stronghold by itself.
Can it find strongholds without throwing eyes?
Not from the seed: that would need the whole world generator. It can say where the rings are, so if you know nothing else, the first three strongholds are 1,280 to 2,816 blocks from 0, 0, about 120 degrees apart.
Is anything sent anywhere?
No. Everything is worked out in this page, on your device. The throws are kept in the page's address so that Copy link shares them.