Besides the mathematical explanation, you can also think about this intuitively using the physics behind it. First, when you increase the volume of the tube, the weight of Hg increases; however, the other consequence of increased volume is increased area. This means that the pressure on the larger mass of Hg is also directly increased with volume of Hg because there is more surface area for the air to push. Second, lets make an analogy to pressure - pressure is the result of molecules hitting the surfaces of an object. You can translate the molecules into rain droplets. When it rains, meteorologists say it rained 5mm, 20mm, etc. How do they know this? They take a flask and expose it to the rain. The height of the water in the flask is used to determine the amount of rain. Here you might ask what about the size of the flask? Well, just like in the case of pressure, the diameter of the flask doesn't matter. Whether you used a test tube with with half an inch diameter or a beaker with 10 feet diameter, the height of the collected rain is going to be the same. You can think of this height as pressure because it is the accumulation of rain droplets that generated the pressure force at the bottom of the flask over time. Is the volume collected here the same? No. It will be much larger - equivalent to a larger pressure force on the larger surface area.
