There is competition between individual members of a species, too. Limited numbers of mates means limited reproduction opportunities. There has to be some competition.
That said, highly social animals like humans, other apes, dolphins, etc. all tend to be altruistic, because you need that trait to live successfully in a cooperative group with others. Living in a cooperative group has strong advantages. You work together to get more food, gain access to resources you otherwise wouldn't have, have regular access to mates, and protect members of the group, particularly offspring. By giving up something (you could just eat all the food you find yourself), you gain the benefits of the group.
People who believe selfishness is a virtue should go live in the woods without anyone's help and see how well they do.