01What crop factor is, in one paragraph
A lens projects a circular image. A sensor sits behind it and records a rectangle out of that circle. A smaller rectangle records less of the circle, so the picture looks more zoomed in even though nothing about the lens changed. Crop factor is just the ratio of the full-frame diagonal to that format's diagonal, and multiplying your focal length by it tells you which full-frame lens would have framed the same shot. A 50mm on Micro Four Thirds frames like a 100mm on full frame. The lens is still a 50mm; it is still gathering light like a 50mm; it just sees less.
02The part that causes all the arguments: aperture
Two true statements that sound contradictory. First: f/2.8 is f/2.8 on every format. Your light meter does not know or care what sensor is behind the lens, and the exposure is identical. Anyone telling you to "correct" your exposure for crop factor is wrong.
Second: for the same framing, a smaller sensor behaves like a slower aperture for depth of field and for noise. Multiply both the focal length and the f-number by the crop factor to get the full-frame combination that produces the same photograph, in both framing and background separation. So 25mm f/1.8 on Micro Four Thirds gives you the picture a 50mm f/3.6 gives on full frame. Both statements hold at once because they answer different questions: one is about exposure per unit area, the other about total light collected across the whole frame.
03What the area column is really telling you
The single most useful number in the table above is area relative to full frame, because sensor area is what drives low-light performance far more than megapixel count does. A full-frame sensor gathers about 2.3 times the light of an APS-C one and four times that of Micro Four Thirds at the same framing and aperture, which works out at roughly one and two stops of noise advantage respectively. That is the honest version of the "full frame is better in low light" claim, and it is worth about one ISO stop per stop of crop.
It cuts the other way too. Landscape and wildlife photographers often want more depth of field, not less, and a Micro Four Thirds body hands them two extra stops of it for free, in a body and lens combination that weighs half as much. A 300mm f/4 on Micro Four Thirds frames like a 600mm and fits in a jacket pocket. There is no format that wins everything.
04Frequently asked questions
No. f/2.8 at 1/200 and ISO 400 is the same exposure on every format. Crop factor affects framing, depth of field and total light gathered across the frame, never the meter reading.
About 2.3x the surface area against a 23.5x15.6mm APS-C sensor, and about 2.6x against Canon's 22.3x14.9mm version. That is roughly one and a quarter stops of light-gathering advantage at the same framing.
Historical manufacturing decisions, nothing more. Canon settled on 22.3x14.9mm for a 1.6x crop while Sony, Nikon, Fujifilm and Pentax use about 23.5x15.6mm for 1.53x. It is a small difference but it changes every equivalence number, which is why both are listed separately here.
Digital medium format, as in Fujifilm GFX and Hasselblad, is 44x33mm: about 1.7x the area of full frame. Real, but far short of the 6x7cm film frame the name evokes. The step from APS-C to full frame is a bigger jump than the one from full frame to digital medium format.