William Hamilton first published his theory of kin selection in 1963 and 1964. The best way to understand the importance of kinship is to take a gene's eye view of evolution and natural selection. For while natural selection acts on individuals (i.e. it is individuals that die or reproduce), it is the genes that are being preserved. The gene is the unit of selection, not individuals. If you look at Natural Selection from this perspective, it opens up the possibility that there could be selection for genes that ensure their own replication even at the expense of the individual.
Thus, through Hamilton's theory of kin selection, the concept of the selfish gene was born. What Hamilton did was to formalize the concept in a way that could be quantified and measured. And like many revolutionary concepts in science, its elegance lay in its simplicity.
Hamilton's formula predicted that selection will favour altruism (i.e. genes for altruistic behaviour will increase in frequency) when:
k > 1/r
Benefit to the recipient = k
Cost to the actor
r = Wright's coefficient of genetic relatedness