Plato, thankfully, did not poison himself; Socrates did. This is described in Plato's dialogue Phaedo. Socrates was a historical figure, whose writings do not survive, and he was famously the teacher of Plato. There are also other inaccuracies in the answer here.
Below are the four arguments for the immortality of the soul as articulated by Socrates in the Phaedo by Plato (from my lecture notes). While one can always make counter-arguments, these are rigorous arguments and ought not be mischaracterized or dismissed without careful consideration:
Argument from Opposites/Cyclical Argument (70b-72a)
1.Those that have an opposite come to be from their opposite. (71a)
2. Being alive is the opposite of being dead.
From 1 and 2:
3. Living creatures come to be from the dead.
4. If something A comes to be something else B then there is a process of becoming from B to A. (71a-b)
From 3 and 4:
5. There is a process of becoming from being dead to being alive. (71d)
6. If there is a process of becoming from being dead to being alive, it must be the process of the dead coming to life. (72a)
7. If something comes to life, it must have existed before birth. (72a)
8. If the process is not cyclical, the same things being reborn as have died, then all things would end up dead. (72a–b)
Therefore, the souls of things must exist when not alive, and undergo the processes of dying and coming to life.
Argument from Recollection (73c - 76d)
1. If a person is reminded of anything, he must first know that thing at one time or another. (73c)
2. Definition: Recollection is knowledge that comes about in this way: when a person upon seeing one thing not only becomes conscious of it, but also of something else which is a different object of knowledge. (73c 5-10)
3. We know that there are such things as absolute Equality (i.e. Forms). (74a)
4. We get this knowledge of these Forms from seeing particular things which are quite different from forms and absolute qualities. (74b - 74d)
From 1, 2, 3, and 4
5. This knowledge is knowledge by recollection. (74d)
6. We must have had previous knowledge of the forms and absolute qualities. (74e)
7. We must have had knowledge of absolute qualities (forms) before the first time we were reminded of them by perception. (75a 1-2)
8. It is impossible for us to have gained this original knowledge of absolute qualities in any other way than through recollection prompted by the senses. (75a 4-7)
9. Before we begin to see or hear we must have somewhere aquired this knowledge of absolute qualities. (75b 3-7)
10. We begin to see and hear at birth. (75b 9-10)
From 9 and 10:
11. We acquired our knowledge of forms before birth. (75c4)
Therefore, our souls existed before we were born.
Affinity Argument (78b - 80d)
1. In order for something to dissolve it must be composed of discrete parts. (78c)
2. Things with discrete parts do change. (78d)
3. Forms do not change. (78c-d)
4. Forms are invisible. (79a)
5. Sensible things are visible. (79a)
From 2, 3, 4, and 5:
6. Invisible entities do not change, while visible entities do change.
7. The soul resembles the invisible and the body the visible (79c)
From 1, 2, 3, 6, and 7:
8. The soul does not have parts and does not change. (79e)
Therefore, the soul is indissoluble.
Final Argument (100a-107a)
- When in us, a Form ‘F’ will never admit its opposite (e.g. Snow never admits hot; two never admits odd) (104 b)
- If something brings the Form into that which it occupies, it will not admit the opposite of the form. (105a)
- The soul always brings along Life into a body. (105c)
- Life and Death are opposites. (105d)
From 2, 3, and 4:
5. The soul will never admit Death.
6. What never admits Death is deathless. (105e)
From 5 and 6
7. The soul always brings along the Deathless.
8. The Deathless is indestructible.
9. What always brings along something indestructible is itself indestructible. (106b)
Therefore, the soul is indestructible.
Plato makes an argument for the "soul" being immortal. Interestingly, he does this very shortly before he poisons himself to avoid being executed by the state for not accepting the local gods and corrupting young people with his ideas.
His argument is a little hard to understand for most people, and I'm not entirely sure it's very effective (but that's another question alltogether!) It has four points that I will try to give a short explanation of:
- The Cyclical Argument -- The essense here is that, like sleeping and waking, "life" for the soul came into being from its opposite, death. The soul existed somewhere before birth, even if we're not aware of it, and once it leaves the human body it will go back to that place. He says, basically, that you were "dead" before being born, and since everything comes from something, you being born created life from this death. The soul, after leaving the body, will just return to that state without being destroyed. It's a strange sounding argument but amounts to "the dead" being born from living things (when they die,) and "the living" being born out of the dead (upon birth.) Either way, the soul existed in both forms.
- Theory of Recollection -- Socrates tells Cebes that people sometimes have knowledge of things that they have never been exposed to and never had the chance to learn. This, he says, is the result of having learned it in some previous life and is proof the soul does not perish.
- The Affinity Argument -- The body is mortal and can die, but the soul is divine and must live. When a person dies they are still around in the form of a corpse that can been seen. They don't simply "poof" away. Similarly, Socrates says that death doesn't simply cause the soul to destruct. He goes further, though, and says that if a person dies smart and balanced and understanding these things his soul will continue on in the underworld in a stable fashion. If, though, the person who has died led a greedy and self-centered life, their soul will not be able to detach from the body that gave it so much pleasure and will be a miserable soul (perhaps even tortured in Hades.) This argument is hard to prove and even Cebes doesn't buy into it for its lack of logical proof.
- Form of Life -- This goes to the idea of "forms." A good way to think of it, according to Socrates, is the number three. Whether people are there to see it or not, there will always be the number three (and all numbers, for that matter.) You can't make three things into four without adding or taking away. That makes different numbers "forms" that are universal. Socrates argues that the soul, too, is a form, and that as a form it cannot be destroyed. Three never becomes two and the existing soul can never become a non-existing soul.
These are some famous arguments that have shaped a lot of people's thinking over the years, but it's important to note that Socrates doesn't perform any slam dunks here (at least in my mind.) Much of his "proof" are logical but require a person to accept the principles he's basing the arguments on (such as the fact that people sometimes remember things they've never learned.) It's all very interesting, but to me a bit unconvincing.