You can understand it in this way, when a car out of oil so it needs to add oil then the car have the power, can continue to drive. When you say"加油“ to someone, actually you give them power and encouragement.

“添油加醋”，说的是某人形容一件事无限制的夸大。

as meaning of "support someone", it has two ways to use: 1) use it alone as an exclamation. In this case, it's hard to translate it. Have you ever watched any NBA game? Fans will yell "defense!!!" when the team supported by the fans should defense. In chinese, 加油 is just like "defense!!!" here. For example, when you are watching some matches, you can cry "加油" to the player that you support. You cry that because you support him and you want him to win. Some people translate 加油 in to "come on!!!" because when people watch sport games, they may cry "come on!!!". Anyway, I think you get what I mean. As I said, knowing how to use it, that's enough, no need to translate it in this case. 2) 加油 can be used as a verb. For example, your friend will pass an exam or a match tomorrow, today you meet him and say: 明天的考试 / 比赛要加油。 or 你要加油。 etc. Seriously, it's hard to translate either... lol... These sentences here mean that you wish your friend a good result. In english, we say: good luck. or some "bad" phrase such as "I hope you can f**k that exam". In chinese, we say: 你要加油 or just 加油 or 考试要加油 etc. Just like the first case, we dont need to translate it because it's almost impossible. If we know when and how to use it, that's enough.

When you say same thing, you mean "add oil" or "supporting somebody"? add oil : 这辆车需要加油了。 support someone : 加油，我们相信你可以的。

Nope,"jia you" means "come on!" It can cheer somebody up,and support to somebody.