Just prior to his speech, Socrates disputes Agathon's claim that Love is beautiful and good:
There is some x such that Love loves x or there is no y such that Love loves y
It is not the case there is no y such that Love loves y
Hence, there is some x such that Love loves x
If there is some x such that Love loves x, then Love desires x
If Love desires x, then Love does not possess x
Love loves/desires what is beautiful
Hence, Love does not possess what is beautiful
If Love does not possess what is beautiful, then Love is not beautiful
Hence, Love is not beautiful
Love loves/desires what is good
Hence, Love does not possess what is good
If Love does not possess what is good, then Love is not good
Hence, Love is not good
Hence, Love is neither beautiful nor good
(1) is plausible by law of excluded middle. Agathon grants (2), so (3) follows. Agathon grants (4) as well, presumably since loving is plausibly understood as a species of desiring. Socrates argues for (5) by observing infelicities, e.g. a bald man desiring to be bald, and accompanied by an explanation of putative counterexamples, e.g. what the bald man desires is that he continue to be bald. Agathon claimed (6) and (10); note I've collapsed the link between loving and desiring from (4) when characterizing these premises. (7) and (11) follow. Socrates motivates (8) and (12) by shifting from possession of an object with a quality to being an object with a quality. (9), (13), and (14) follow.
Why accept (4)? To be fair, I think it's right to say that if S desires x then S does not possess x, so I'm happy to grant the related (5). Desire seems motivational, and so intimately tied to action. If S desires something S already possesses, there seems little motivation or guidance for action on offer unless one appeal to something like continued possession, as Socrates points out. That said, (4) treats loving as a species of desiring. But it doesn't seem as obvious that, say, loving is motivational. S might love x without that love motivating or guiding action, e.g. love of an ancestor, love of a mathematical proof. Yet, this must be the case if loving is a species of desiring.
Why accept (8) and (12)? Socrates seems to shift from the lack of possession of an object to the lack of having whatever quality that object exhibits possesses. If this conditional is true, it's nevertheless irrelevant, since there seems little connection between, say, my not possessing a red apple and so thereby not being red. That is, it seems plausible Love might lack beautiful things, yet still be beautiful. Moreover, Love might seek out beautiful things because Love is beautiful, if one assumes - as many of the speakers seem to - that like attracts like.