Why should a person who disbelieves after becoming Muslim be executed?
Question

                The punishment for the apostate is execution. Why such harshness?.
Praise be to Allah.
    The punishment for apostasy from the religion of Islam is 
    execution. Allaah says (interpretation of the meaning): 

    “And whosoever of you turns back from his religion and 
    dies as a disbeliever, then his deeds will be lost in this life and in the 
    Hereafter, and they will be the dwellers of the Fire. They will abide 
    therein forever”

    [al-Baqarah 2:217]
And it was proven that 
    the Prophet  (peace and blessings of Allaah be upon him) said: “Whoever 
    changes his religion, execute him.” Narrated by al-Bukhaari in his Saheeh. 
    What this hadeeth means is that whoever leaves Islam and changes to another 
    religion and persists in that and does not repent, is to be executed. It was 
    also proven that the Prophet  (peace and blessings of Allaah be upon 
    him) said: “It is not permissible to shed the blood of a person who bears 
    witness that there is no god but Allaah and that I am the Messenger of 
    Allaah except in three cases: a life for a life, a previously-married person 
    who commits adultery, and one who leaves Islam and forsakes the jamaa’ah.” 
    Narrated by al-Bukhaari and Muslim. 

    This harsh punishment is for a number of reasons: 

    1 – This punishment is a deterrent to anyone who wants to 
    enter Islam just to follow the crowd or for hypocritical purposes. This will 
    motivate him to examine the matter thoroughly and not to proceed unless he 
    understands the consequences of that in this world and in the Hereafter. The 
    one who announces his Islam has agreed to adhere to all the rulings of Islam 
    of his own free will and consent, one of which rulings is that he is to be 
    executed if he apostatizes from the faith. 

    2 – The one who announces his Islam has joined the jamaa’ah 
    (main body) of the Muslims, and whoever joins the main body of the Muslims 
    is required to be completely loyal and to support it and protect it against 
    anything that may lead to fitnah or destroy it or cause division. Apostasy 
    from Islam means forsaking the jamaa’ah and its divine order, and has a 
    harmful effect on it. Execution is the greatest deterrent that will prevent 
    people from committing such a crime.

    3 – Those Muslims who are weak in faith and others who are 
    against Islam may think that the apostate has only left Islam because of 
    what he has found out about its real nature, because if it were the truth 
    then he would never have turned away from it. So they learn from him all the 
    doubts, lies and fabrications which are aimed at extinguishing the light of 
    Islam and putting people off from it. In this case executing the apostate is 
    obligatory, in order to protect the true religion from the defamation of the 
    liars and to protect the faith of its adherents and remove obstacles from 
    the path of those who are entering the faith. 
4 – We also say that the 
    death penalty exists in the modern laws of man to protect the system from 
    disorder in some situation and to protect society against certain crimes 
    which may cause its disintegration, such as drugs etc. If execution can 
    serve as a deterrent to protect man-made systems, then it is more 
    appropriate that the true religion of Allaah, which Falsehood cannot come to 
    it from before it or behind it  [cf. Fussilat 41:42], and which is all 
    goodness, happiness and tranquility in this world and in the Hereafter 
    should punish those who commit acts of aggression against it and seek to 
    extinguish its light and defame its image, and who fabricate lies against it 
    to justify their apostasy and deviation.

     Fataawa al-Lajnah al-Daa’imah, 21/234-231.