The only way to simulate with a camera and lens what our brain does when we look at distant objects is to use a much longer focal length. This, of course, also reduces the field of view in a dramatic way.

If you want a foreground object to remain the same size in your photo while increasing the size of objects in the background you can back away from the foreground object and then use a longer focal length to maintain the size of the foreground object.

For example, if you are 20 feet from the foreground object with a 50mm lens you can back up to 40 feet and use a 100mm lens. You've double the distance to the foreground object (increased it 100%) as well as doubled the focal length, so the foreground object should appear roughly the same size assuming it is relatively flat. Items 1,000 feet behind the foreground object will appear almost twice as large: you've only increased the distance to the background by just under 2% from 1,020 to 1,040 feet, yet you've doubled the focal length! And you'll only capture about 1/4 as much of the background scene with a 100mm lens than with a 50mm lens.

This is more a question about human physiology and psychology than about photography. Our brains have an amazing capacity to "zero in" on a particular portion of our vision's field of view and ignore much of the rest of the information coming from our optic nerves. The same is true of our auditory nerves that allows our brains to filter background noise. This is also related to why the moon looks much larger near the horizon than when higher in the sky. When the moon is closer to the horizon as a point of reference we block out more of the surrounding scene than when it is further from it.