Let me try this angle:
The actual, physical size of the FX and DX sensors are different:
This is readily apparent if you look at an FX body next to a DX body, and can see the actual sensors when both are set to "Mirror Up for Cleaning":
So how does this translate into anything meaningful out in the 'real world'? Well, lets' pretend we're out there taking photos. And we come across this peaceful scene:
Now, an FX lens on an FX body has to project a large enough image to cover the larger FX sensor, so it will project an image into the camera that looks like this:
(Yeah, I know.... it's right-side-up. In reality, the image would be upside-down, but let's ignore that for the purpose here.)
The lens needs to create a large enough image to cover a sensor measuring 24x36mm (represented by the white rectangle):
So an FX lens/body will record the final image as:
Now let's take the same focal length lens, but only it's a DX-format lens. It will project a smaller circle:
Because it only needs to cover a sensor that measures 18x24mm:
So the same focal length lens, on a DX body, will record a final image as this:
Now, if we put the two final images (FX and DX) side by side, we end up with this:
Notice how the subjects in both images are exactly the same size? It's just that the DX sensor recorded a smaller portion of the scene because the sensor is physically smaller! This results in a narrower field of view.
So if you compare the sensor size:
with the above two images, you should be able to understand the 'crop sensor' effect on field of view.