One big reason for this is because individuals of the same species often have to fight over shared resources (this means shared food and shared environment).
Because they fight over a limited amount of space and food, competition between individuals within one species can be present and very tough.
On the other hand, having members of different species also possibly means different territories and different food sources. A lion has no quarrel with dolphins, while it can have a lot of fights with lions from different prides (large group of lions).
