One small addition to Lorenz' explanation of the 'kinship' part of the Affinity Argument: Socrates' grounds for claiming a kinship between the soul and the realm of the invisible and eternal, are not located entirely in the Affinity Argument, but they are als0--and perhaps largely--given in an earlier passage, called 'Socrates' Defense' (64a-69e). This is the passage which immediately precedes the Cyclical Argument.
In the Affinity Argument, Socrates refers back to this 'Defense' at 79c2 and d9:
Although 'Socrates' Defense' is not usually counted among the passages in the Phaedo that give an 'argument' for immortality, it's there that Socrates gives his fullest account of how the soul behaves when it operates 'alone by itself'.
ou)kou=n kai\ to/de pa/lai e)le/gomen, o(/ti h( yuxh/, o(/tan me\n tw=| sw/mati prosxrh=tai ei)j to\ skopei=n ti h)\ dia\ tou= o(ra=n h)\ dia\ tou= a)kou/ein h)\ di' a)/llhj tino\j ai)sqh/sewstou=to ga/r e)stin to\ dia\ tou= sw/matoj, to\ di' ai)sqh/sewj skopei=n ti to/te me\n e(/lketai u(po\ tou= sw/matoj ei)j ta\ ou)de/pote kata\ tau)ta\ e)/xonta, kai\ au)th\ plana=tai kai\ tara/ttetai kai\ ei)liggia=| w(/sper mequ/ousa, a(/te toiou/twn e)faptome/nh;Now we have also been saying for a long time, have we not, that, when the soul makes use of the body for any inquiry, either through seeing or hearing or any of the other senses--for inquiry through the body means inquiry through the senses,--then it is dragged by the body to things which never remain the same, and it wanders about and is confused and dizzy like a drunken man because it lays hold upon such things?
pote/rw| ou)=n au)= soi dokei= tw=| ei)/dei kai\ e)k tw=n pro/sqen kai\ e)k tw=n nu=n legome/nwn yuxh\ o(moio/teron ei)=nai kai\ suggene/steron;And now again, in view of what we said before and of what has just been said, to which class do you think the soul has greater likeness and kinship?