Why is man mortal?
We are not mortal. In fact no living being in true sense is mortal. it is only owing to our ignorance that we think that we are being born, growing, dying etc.
in Bhagavad Geeta Krishna Says:
As the embodied soul continuously passes, in this body, from boyhood
to youth to old age, the soul similarly passes into another body at
death. A sober person is not bewildered by such a change.
It is only body that dies at the time of death but soul is imperishable(BG 2.20). Our body is constantly changing at every moment. At some point we had a body of infant; than a boy, .... So in this process we can see that our body is changing but we feel like same person.
So what is it that has not changed? What is that which animates this body?
A simple logic may be applied. No machine in this world works without a touch of living entity so how can this machine of body work independently? ex If some one thinks that the aeroplane flies on its own; then such a person will be called fool. that is because we know that aeroplane is simply made out of dead matter and so is our body; and we know that dead matter cannot move.
But intelligent man will think; just like a pilot in aeroplane; there must be some one inside that moves this body and when that 'some one'(Soul) leaves the body; this body becomes useless.
Where Does the soul go after quitting current body?
krishna says in BG 8.6
Whatever state of being one remembers when he quits his body, O son of
Kuntī, that state he will attain without fail.
So whatever desires a person has at the time of death; soul gets a body that is fit to fulfill those desires.
Just like how air carries aroma so does subtle body(mind and intelligence) take the soul to another body.
So in this way soul transmigrates from one body to another untill he comes to the true knowledge and is freed from all desires and duality of this world.
What is the reason for desires?
Krishna says in BG 3.37
The Supreme Personality of Godhead said: It is lust only, Arjuna,
which is born of contact with the material mode of passion and later
transformed into wrath, and which is the all-devouring sinful enemy of
So it is lust only that gives rise to unlimited desires. Lust here means mentality to enjoy selfishly or to simply engage in sense gratification out of ignorance.
Why We have to take birth inspite of doing good Karma?
Inspite of doing pious activities such as charity etc; still person has to take birth and die again and again. This is because good karma gives you pious credits and hence you have to come in this world again to enjoy the good karma you have earned and in case a person commits sinful activities he has to come to suffer the reactions.
Nevertheless both type of activities bind the doer to this material world unless he cuts the thread of attachment with the sword of knowledge and engage himself in service of Supreme Lord.
Why does the soul Come to this mortal world?
originally we are all the spiritual energy of God(souls). Soul is by nature full of eternal bliss and knowledge. The spirit soul however has minute degree of independence. When this indepence is properly used. it is used in service of Supreme lord.
However sometimes due to this minute independence soul wishes to lord it over the things. i.e. he becomes envious of God. We can see that all around. people drowned in ignorance wants to become the master of their family, state country etc. Originally Krishna(or Vishnu) is the master of everything(BG 5.29).
A person in full consciousness of Me, knowing Me to be the ultimate
beneﬁciary of all sacriﬁces and austerities, the Supreme Lord of all
planets and demigods, and the benefactor and well-wisher of all living
entities, attains peace from the pangs of material miseries.
So for such envious souls; they are put in this world. This world is just like a prison house for such envious souls. Anyone(as you state in your question) who doesnt want to get out of this prison house is under the spell of illusion and hence must transmigrate from one body to another. This world is actually very miserable but owing to our ignorance we find it very nice. Even in Geeta Krishna says(BG 8.15) that it is a miserable place for living entities.
After attaining Me, the great souls, who are yogīs in devotion, never
return to this temporary world, which is full of miseries, because
they have attained the highest perfection.
What is the reason for repeated birth and death in this material world?
It is only desire for personal sense Gratification that binds the soul to a chain of repeated birth and death.
Basically God respects our free will; and if out of sheer ignorance , someone wants to carry on in this world then that wish is fulfilled.i.e. whatever wish a person has at the time of quitting current body; he gets the next body based on that. This is the reason Krishna says at the end of Geeta
Thus I have explained to you knowledge still more conﬁdential.
Deliberate on this fully, and then do what you wish to do.
He doesn't force Arjuna to fight the battle.Instead it leaves it fully up to him.If God did not respect the free will of us He would not have said that.
How to get out of this world?
krishna makes it very clear in Bhagavad Geeta.
Abandon all varieties of religion and just surrender unto Me. I shall
deliver you from all sinful reactions. Do not fear.
Lord makes it clear that by surrendering unto Him ; we will be delivered from this samsara. When a person is fully situated in Knowledge of supreme and has transcended all the dualities of this world; he finally comes to the point of total surrender to God.