This is related to relative velocity.I get that, from my moving frame of reference rain is making an angle.But still... it doesn't make sense to hold umbrella at an angle when rain is falling vertically down.
I guess you're asking for an intuitive explanation, because it seems to be mathematically clear.
Well, imagine a set of three columns of raindrops, each row containing three drops. Let's call columns A B C and rows 1 2 3, because they fall orderedly. You first encounter drop 1, then the second, then the third.
For a standing pedestrian, he will get hit by three drops: A1, A2 and A3, that way.
However, if you walk rightards, you first get drop A1, but then, while you walk forwards, row 2 has fallen down, so you will encounter drop B2, and then, for the same reason, you'll find C3. Those drops have been circled in the picture.
So, what you see is "like" if the rain were falling diagonally. The only way for those drops to find you is that the drops "tilt" towards you, but that's just your perception. A standing man will see vertical rain. It all depends on the reference frame.
Imagine yourself in a train car with no roof (to let the rain in) and no windows (so you don't know you're moving). In that car, all you know is that the rain is coming at you at an angle to the vertical. That the rain is falling vertically to someone standing by the train tracks is not relevant to your experience. To best prevent from getting wet, you will hold the umbrella at an angle to the vertical.
There are other observers moving in different directions and at different speeds relative to the train tracks. In their frames of reference, they all experience the rain falling at different angles. When you say "the rain is falling vertically down" that is just referring to one particular frame of reference and no one frame has the "correct" or "actual" direction of the rain.
The main reason is that in the frame of reference of the person, the rain will be seen to be falling at an angle. Perhaps a more intuitive way of thinking about the problem is by considering the moment when a raindrop is right inches/centimeters in front of the person's eyes. If the person is just standing, the raindrop will not hit the person, but if the person is walking forward then the raindrop will hit the person because of this relative angle at which the rain falls from the perspective of the person.
A related example of the same effect of reference frames is what happens when someone tries to swim directly across a fast-moving river. In order to make it to the other side, the swimmer should travel at an angle to counteract the effect of the river's speed.