At the conclusion of Agathon's speech there is general applause, and everyone remarks on how eloquent and brilliant Agathon is. Socrates points out to Eryximachus that what he had said before (at 194a) has held true, that Agathon would speak admirably and that he would be lost for words. Eryximachus agrees that Agathon has spoken beautifully, but does not doubt that Socrates will find something to say.
Socrates expresses great admiration at the beauty and careful phrasing of Agathon's speech, noting a debt of influence to the great orator, Gorgias. Socrates confesses that he has realized that he has no business in trying to eulogize Love, since clearly he knows nothing at all about how to give a proper eulogy. He had thought that the best way to give a eulogy would be to tell the truth about the subject and to present one's subject in the best light the truth can shed. In fact, he says, it seems that the truth does not matter at all, and that one should simply ascribe all the finest possible qualities to one's subject whether or not what one says is true. All the speeches given so far have been beautifully expressed and have said all sorts of marvelous things about Love that might impress the ignorant, but none of them have approached the truth.
Socrates admits that he would be incapable of following up Agathon's speech with a similar eulogy, and if that is his task, he must bow out. However, he asks Phaedrus if he might be permitted to give a different kind of eulogy, one that speaks the truth and doesn't use fancy language, but uses only the words that come to him as he speaks. Phaedrus consents to this request, and consents also to Socrates' further request that he might ask Agathon some questions so as to make his speech on the basis of an agreement with the tragedian.
Socrates begins by asking Agathon whether or not Love is love of something or other, in the same way as a father is a father of a son or a daughter or the way a brother is the brother of a sister or a brother. Agathon agrees that Love is certainly love of something, which prompts Socrates to secure Agathon's further agreement that Love desires what it is love of. If Love desires what he loves, that would suggest that, necessarily, he does not have the object of his desire in his possession. When we desire something we have--for instance, a healthy person who wants to be healthy--what we desire is to continue to have that thing in the future, not in the present.
Socrates then recalls what Agathon said that Love is of: for one thing, Agathon suggested that the gods are organized through a love of beautiful things. From this, Socrates has Agathon agree that Love must be love of beauty, which in turn implies that Love itself must be wholly without beauty. Agathon admits that Socrates is right and that he was wrong. Socrates goes on to point out that if good things are beautiful, then Love must also be lacking in good things, and cannot himself be good. Agathon surrenders, saying that he cannot argue against Socrates. Socrates replies that it is easy to argue against Socrates, but that Agathon cannot argue with the truth.
This passage is the only example we find in the Symposium of the method of elenchus, or cross-examination, that defines Plato's and Socrates' dialectic method. In a typical dialogue, Socrates will question someone who claims expert knowledge on a particular subject. Supposedly without interjecting any knowledge or thoughts of his own, Socrates leads his interlocutor into a contradiction, showing him that his supposed knowledge is unfounded. Here Socrates takes Agathon's claims to know the nature and importance of Love, and leads Agathon, through questioning, to admit that he does not know the subject well at all.
You do not have to look outside you as everything is in your "self", everybody is the two energies (woman-man/yin-yang/...) and if you awaken your "opposite" you are "one", you are realized, totally happy. This is called e.g. in the Bible the Holy or Mystic Marriage. Also the alchemists describe it as "Coniunctio Oppositorum", the Philosopher's Stone, make Gold from Lead,... See the writings of the Mystics too. Etc.