Sandman, tenfour, and Mag Roader make good points about using a real plane's joystick and controlling the motion of the head, rather than the eyes.
You can also think about if you are walking in a busy space - say a busy workplace where many people must walk through a small confined space, but each must hurry to complete their tasks. If you've just finished carrying something to its destination and quickly turn around to walk back to where the rest of your load is, but there is someone who you didn't see or hear walking up behind where you were just standing, your immediate reflex is to pull back to not collide with them. If you were distracted in an airplane and suddenly realized that you were about to fly into an obstacle (other pilot, birds, projectiles, mountain, tree, ground) your reflex would inspire you to perform this same motion: "Pull back!". It makes sense, then, that aircraft controls would mirror this instinctive human response. You can even think of sitting in a cockpit and wanting to look down at what is in front of you, but moving underneath your field of view. In any other situation, you would want to lean forward to peek over the edge of your supporting platform, so leaning the joystick forward to look downward over the nose of the aircraft is also natural.
One thing that I think deserves to be drawn out is that we talk about 'Up' keys or 'Up' on a Directional-pad, but almost never do we actually position our keyboards or controllers/joypads so that the so-called 'up' cursor key actually points up. Usually, a keyboard lays flat on our desk and a controller lays face upwards in our hands, perhaps only slightly tipped back toward us. Taking this into account, once you've held an aircraft's joystick in your hand in real life, it is very easy to stop thinking of the 'Up' key as 'Up', and rather as 'Forward'. Likewise, the 'Down' key ceases to be 'Down' and becomes, simply, 'Back'. This is a small transformation in thinking, but actually usually more accurate to how we use our keyboards, joypads, and mice, than mentally translating their directional inputs 90 degrees in space.