What is the working principle of inclined tube manometer?

For a very basic explanation, a manometer measures the pressure differential between a closed pressure applied at one end and an open pressure at the other.

For example. If you have a typical u-bend manometer and you apply an air pressure at one end, the liquid will raise correspondingly, as the pressure on the closed side increases more and more above the ambient air pressure.

The amount it rises depends on the density of the liquid used and the diameter of the tubing.

So basically a manometer uses the height of a liquid to translate the amount above ambient air pressure, which is considered zero, of your pressure source.