why is the world round?

If you stand on the seashore and watch a ship sailing away, it will gradually disappear from view. But the reason cannot be the distance: if a hill or ower are nearby, and you climb to the top after the ship has completely disappeared, it becomes visible again. Furthermore, if on the shore you watch carefully the way the ship disappears from view, you will notice that the hull vanishes first, while the masts and sails (or the bridge and smokestack) disappear last.
It is as if the ship was dropping behind a hill, which in a way is exactly the case, the "hill" being the curve of the Earth's surface.

"why does the world go round?"

The Earth is rotating around an axis (called its rotational axis). Some objects rotate about a horizontal axis, like a rolling log. Some objects, such as a skater, rotate about a vertical axis. The Earth's axis is tipped over about 23.5° from vertical. How do we define up and down in space? What would "vertical" mean? For the Earth, we can think of vertical as straight up and down with respect to the plane in which the Earth orbits the Sun (called the ecliptic). Earth's rotational axis points in the same direction relative to the stars, so that the North Pole points towards the star Polaris. Think of the Earth as a spinning top, tipped over to one side. Over very long time periods (thousands of years) the direction of Earth's axis slowly changes due to precession. The Earth rotates around once in 24 hours - that's a rate of 1000 miles per hour!. The time it takes for the Earth to rotate completely around once is what we call a day. It's Earth's rotation that gives us night and day. The combined effect of the Earth's tilt and its orbital motion result in the seasons.

"what makes you feel that the world is not round?"

"when did man actually come to know that the world is round?"

It is conjectured that the first person to have advocated a spherical shape of the Earth was Pythagoras (6th century BC), but this idea is not supported by the fact that most presocratic Pythagoreans considered the world to be flat. Nobody is sure who first discovered that the world is round. It is most likely that it was done by observing the Earth's shadow on the Moon during a lunar eclipse. Aristotle (384-322 BC) said that it was common knowledge, at least among the learned, so it's been known for at least 2,500 years.
We do know that Eratosthenes of Cyrene was the first to accurately estimate the Earth's diameter, around 220 BC. How he did it is detailed at Astronomy Online.

Today it is well known that the Earth is a sphere, or very close to one (its equator bulges out a bit because of the Earth's rotation). When Christopher Columbus proposed to reach India by sailing west from Spain, he too knew that the Earth was round. India was the source of precious spices and other rare goods, but reaching it by sailing east was difficult, because Africa blocked the way. On a round globe, however, it should also be possible to reach India by sailing west, and this Columbus proposed to do (he wasn't the first one to suggest this--see below).
Sometimes the claim is made that those who opposed Columbus thought the Earth was flat, but that wasn't the case at all. Even in ancient times sailors knew that the Earth was round and scientists not only suspected it was a sphere, but even estimated its size.