"I have a high temperature."
Translation:Я маю високу температуру.
У мене є is I have smth, I own smth, e.g. bag or dignity. You can hardly say that you own high temperature :) Though, the boundary between these two is rather blurred. To tell the truth, У мене є висока температура is not quite wrong, but sounds weird.
Because there's not a direct object of a verb in there. У мене висока температура is literally "there is a high temperature with me", which means high temperature is the subject and has to be in the nominative.
If you used the verb мати, then you would have a direct object and need the accusative form: Я маю високу температуру.