The other answers don’t seem to be addressing your main question, which is “why does an eclipse start and stop so suddenly, instead of coming on slowly over the course of days?”
The moon’s speed in its orbit around earth is around 1km per second. The moon’s width is 3474 km. That means it takes 3474 seconds, or less than an hour, for the moon to move its own width.
The width of the sun as seen in the sky is approximately the same as the width of the moon. So, to get from one side of the sun to the other, the moon just needs to move one moon-width, which we calculated takes less than an hour.
To calculate the exact time an eclipse lasts from a point of view on earth, you need to factor in other things like your movement on the surface of earth as it rotates and the movement of the earth itself, but the basic answer to your question is “the moon is going really fast, and it doesn’t take very long for it to move one sun-width in the sky, so it can go from not overlapping, to overlapping, to not overlapping again very quickly.”
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