When you stand on the hill, the horizon you see is farther away and so looks higher. What is happening can be illustrated as follows: draw a circle (or trace a drinking glass or whatever). This is the earth. Now, put a little stick standing on the circle (you). Use a ruler and find the farthest point away from you that the straight edge touches the circle. This is as far as you can see. Mark it. Now, draw a hill, which is bigger than you, and stand on top of it, and repeat the process. You will notice that you can see farther.When standing on the shore, you can only see ground(or water) at the same altitude as you are that is about six miles away due to the curvature of the earth. If you're higher up (or just taller), you see farther.

Don't have a really good answer for this other than to say that I think it is difficult to tell if we are really looking horizontally; we don't have a good sensor in our head that tells us what the angle is between where we are looking and some reference such as the zenith or a the line tangent to an Earth radius or something. Another issue is that sometimes when looking out to the horizon it is difficult to tell where the ocean ends and the sky begins but it sounds like this may not be a concern for you.

You should be able to make a simple instrument to measure the angle between the horizon and a line that points towards the Earth's center. You can take a protractor, drill a hole in the "center," tie a string through the hole, and tie a weight on the other end of the string. If you sight along the edge of the protractor towards the horizon, someone else can see what the angle is between the string and the horizon.