The Cyclical Argument, or Opposites Argument explains that Forms are eternal and unchanging, and as the soul always brings life, then it must not die, and is necessarily "imperishable". As the body is mortal and is subject to physical death, the soul must be its indestructible opposite. Plato then suggests the analogy of fire and cold. If the form of cold is imperishable, and fire, its opposite, was within close proximity, it would have to withdraw intact as does the soul during death. This could be likened to the idea of the opposite charges of magnets.
The Theory of Recollection explains that we possess some non-empirical knowledge (e.g. The Form of Equality) at birth, implying the soul existed before birth to carry that knowledge. Another account of the theory is found in Plato's Meno, although in that case Socrates implies anamnesis (previous knowledge of everything) whereas he is not so bold in Phaedo.
The Affinity Argument, explains that invisible, immortal, and incorporeal things are different from visible, mortal, and corporeal things. Our soul is of the former, while our body is of the latter, so when our bodies die and decay, our soul will continue to live.
The Argument from Form of Life, or The Final Argument explains that the Forms, incorporeal and static entities, are the cause of all things in the world, and all things participate in Forms. For example, beautiful things participate in the Form of Beauty; the number four participates in the Form of the Even, etc. The soul, by its very nature, participates in the Form of Life, which means the soul can never die.