Most people grow up in the shadow of at least one of the major religions, all of which suppose some kind of survival beyond death. They therefore usually take it for granted that we can "know what we are talking about" when we speak of immortality or less grandiosely of survival after death. We assume sense has been given to talk of the soul as something correlative with the body, as Plato does when the discussion starts and death is characterized as the beginning of the separate existence of the soul and of the separate existence of the body. Although Socrates has no proof that the soul does exist after the body dies, his arguments can be convincing to many by using the dialectic.
The dialogue between Socrates and the others begins with a longer statement of the type of defense Socrates gave in the Apology for thinking that death would be preferable to life for one with his interests: the desire for wisdom is frustrated by the body in various ways, so one can hope to get much further without the body's impediments. The philosopher's life is then a practicing to die [67-68, 80e-81]. The real argument starts when Cebes objects that the position depends on the assumption that the soul, or we might say, our mental activity aimed at understanding does actually hang together after the death of the body.
The main arguments concerning the immortality of the soul come from the Phaedo. Socrates believed that when his body ceased to exist anymore, that his soul would leave and join that of the forms, where he would be eternally. Socrates believed so strongly in this, that not only did he not fear his death, he welcomed it. He believed that only when the soul separated from the body, is a person able to be truly enlightened and gain all knowledge. This enlightenment has been Socrates" life long goal of discovering the truth. Even at his hour of death, Socrates showed no hesitation. However, Socrates" friends did not believe so strongly, and took some great convincing by Socrates, to allow his friends to be okay with his death.