When a 70-300mm DX lens is used on a DX body, your field of vision will be 1.4x greater than 70-300mm ... so the image will appear as being taken with a field of view of 98-420mm (the comparable viewing range of an FX body). That's because a DX body is a crop factor body. It's field of view is 1.4x greater than what it would be on an FX body.
IF you use the 70-300mm DX lens on an FX body, you should have a settings choice – to be able to choose between what it would look like if using it on an FX body or the DX view where the image already appears cropped by the 1.4x sensor. You will also lose pixels with this last option since the image gets cropped in camera.
Here's the problem ... when a DX lens is used on an FX body, and the camera body is set to record the image as though it was taken with an FX body, you will get a lot of vignetting around the edges of the photos (this will be the option to not have the images cropped in camera). But when setting the FX camera body to use the DX lens as a DX lens, it will automatically crop the image so no vignetting will appear. It's the potential vignetting that causes issues when using a DX body on an FX body. So most people simply have their FX bodies set to automatically adjust the images to be cropped in camera and to record the DX field of view.
Clear as mud, I'm sure. It's all due to a DX body having a field of view that is amplified by 1.4x the view of an FX body.