Aristophanes' speech comes in the form of a myth. Long ago, he explains, there were three genders: male, female, and androgynous, and each person was twice what they are now. That is, they had four hands, four legs, two heads, two sets of genitals, and so on. They could move both forward and backward and would run by spinning themselves around cartwheel-like on all eight limbs. Males were descended from the sun, females from the earth, and those who were androgynous were descended from the moon. They were very powerful and vigorous and made threatening attacks on the gods. The gods did not want to destroy them because they would then forfeit the sacrifices humans made to them, so Zeus decided to cut each person in two. He also suggested that if this didn't settle humans down, he would cut them in two once again and they would have to hop about on one leg.
As each person was cut in two, Apollo turned their heads and necks around so that they would be facing toward the gash that had been made, so as to remind them constantly of the punishment they had been dealt. He also pulled their skin tight to cover up this gash, tying it together at the navel.
Because they longed for their original nature, people kept trying to find their other half and reunite with it. When they found their other half, they would embrace and stay together, not wanting anything else. Eventually, people started dying of hunger or general inactivity. Zeus took pity on them, and moved their genitals around so that they would be facing frontward. This way, when they embraced, they could have sexual intercourse, and those who were formerly androgynous could reproduce, and even two men who came together could at least have sexual satisfaction and then move on to other things. This is the origin of our instinctive desire for other human beings. Those who are interested in members of the opposite sex are halves of formerly androgynous people, while men who like men and women who like women are halves of what were formerly whole males and females. Aristophanes applauds male-male relationships between men and boys since such couples value boldness, braveness, and masculinity, both in themselves and in others.
When we find our other half, we are overwhelmed with affection, concern, and love for that person. This great amount of care cannot result simply from a desire for sex, but we have difficulty articulating precisely what it is that makes us care so much. If Hephaestus, the blacksmith god, were to offer to weld a couple together so that they would become one and never be parted, even in death, they would leap at this opportunity. "Love" is the name that we give to our desire for wholeness, to be restored to our original nature.
Aristophanes observes that if we are disobedient or disorderly toward the gods, Zeus might split us in two once more, so we must strive ourselves, and encourage others, to behave well toward the gods. In this respect, Love is our leader, and if we work against Love we will find ourselves on the wrong side of the gods. Aristophanes urges Eryximachus and the others not to take his speech as a simple comedy, or a joke directed at such life-partners as Pausanias and Agathon. Given that we are all separate, Love does what he can for us given the circumstances: he guides us toward those who are close in nature to us and who best fit our character. Perhaps if we continue to show reverence to the gods, he may one day restore us to our formerly whole selves.
While Aristophanes tells Eryximachus twice that his speech is not intended as a joke, it is not hard to tell that it comes from the imagination of a comic playwright, particularly if we understand "comic" as meaning far more than just "funny." Rather than present a speech of carefully crafted rhetoric, Aristophanes gives us a myth that is wildly imaginative and very entertaining. The story is pleasant enough and has an uplifting conclusion, with the suggestion that Love helps us to find our "other half" and that one day we might be fully reunited.
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.