While preparing for Astronomy Olympiad I came across many problems in which we had to calculate at what time a planet or a star would rise at the Earth, and to solve this many solutions found the angle subtended at Earth by the planet and then correlated it with the fact that Earth covers 360 degree in 24h. Now I did not understand at all why this was done and why is it giving us that time. Could anyone help me with this?
It's a little difficult to follow your question. But if I understand it correctly then you are considering the angle between the planet and the sun subtended at the Earth.
Now suppose that angle is 180 degrees. The the planet is at opposition and the planet will rise at sunset (and set at sunrise) whatever that is in local time.
If the angle is 90 degrees (and recalling that the planet is approximately on the ecliptic plane), then the planet will rise and set at local noon and midnight (or the other way round if the angle is counterclockwise)
If the angle between the planet and sun is $x$ degrees then the planet will rise $x/15$ hours after/before the sun. At least approximately and ignoring thing like inclination of the planet's orbit. The "15" is because the Earth turns 15 degrees in one hour (or 360 degrees in 24 hours).
So the rise and set times, as local solar times, can be estimated from the angle subtended between the planet and the sun.