Focal length

Why on earth does a longer focal length cause a shallower depth of field? I've read it a hundred times, that it's the same as with the distance to subject. How the hell is it the same, when more distance between the sensor and the lens causes a narrower dof, but more distance between the sensor and the subject causes a wider dof??

Minimumfocus distancecalculator

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FOV to focal length

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I know it has something to do with the angle of the light reaching the sensor, I know that. But even if I comprehend that much...

Also, this is my idea of focal length affecting dof: A longer focal length causes the angle of the light rays hitting the sensor to be more (how should I put this...) stretched out, or something like that, fuck, they make the circle of confusion smaller. BUT. Increasing the focal length increases the diagonal length of the aperture too, so the circle of confusion increases in size. BUT ALSO. The longer focal length "zooms into the background" more, thus zooming into an out-of-focus plane and making it more blurry. Is this correct, or I'm just overthinking it?