Of course communists wouldn't say that they claim the individual to be insignificant, but that's what it is, by definition. Please, prove me wrong.

There's two ways to regard life: as an end in itself, and as a means to an end.

The logical conclusions to both are:

End in itself: Each life is of value in and of itself. Nothing external can define the internal value of a life. Each person has the right to live their own life, and no one has the right to lay claim to another person's life(this immediately voids the idea of communism). Each person has the right to their own life, the right to their own mind, and therefore also the right to the product of their own mind, the product of their own life.
The sociel system designed around this ideal: capitalism.

Means to an end: each person's right is only of value in what they serve to some other end - there is a moral obligation to serve that end. Invididuals are insignificant.
Social system designed around this: communism. Communism claims that each person has the moral obligation for "the good of the people." Each person does NOT have the right to their own life, and the product of the lives of those that succeed, those that have the curse of ability, are taken from them by force to those that have not earned it. Ability, success, wealth are punished and mediocrity encouraged. "From each according to his ability, to each according to his need." By claiming need, a person claims a moral right to the product of another's life.