According to verse 6 of surah Ahzab, the wives of the prophet (pbuh) are like the mothers of the believers and have the ruling of their mothers. Consequently, marrying them is haram. Numerous reasons have been mentioned for such, for instance, so that the hypocrites won't take advantage of their status for their own aims, and in respect for the prophet (pbuh) and his spouses.
In verse 6 of surah Ahzab, Allah (swt) says: “The Prophet has a greater claim on the faithful than they have on themselves And his wives are their mothers…”. According to this verse, the prophet’s (pbuh) spouses are like the mothers of the believers and clearly, the believers can't marry their mothers. One thing that must be kept in mind is that since the Quran and Islam are the ones telling us that they are like their mothers, it is also their responsibility to specify the boundaries of this similarity (between their true mothers and the wives of the prophet (pbuh)) and its limitations (how much they are like their real mothers; can they also see them without hijab or are they only like them in the fact that they can't marry them, although all other non-mahram rulings still apply to them). Thus, in verse 53 of surah Ahzab, after expressing how others are supposed act when confronting and speaking with the prophet (pbuh), Allah (swt) says: “…but Allah is not ashamed of [expressing] the truth. And when you ask anything of [his] womenfolk, ask it from them from behind a curtain. That is more chaste for your hearts and their hearts. You may not torment the Apostle of Allah, nor may you ever marry his wives after him. Indeed that would be a grave [matter] with Allah!” (This shows that they are like their mothers only from one perspective, and that is that they can't get married to them).
What this verse is clearly saying is that marrying any of the prophet’s (pbuh) wives after him is impermissible. The reason for such can be the following:
1- If this matter was to be permissible, some could take advantage of it and use being the husband of the prophet’s (pbuh) former wives to their personal benefit and develop a false status in society. There were also chances of them altering and distorting Islam with the excuse that they are aware of what goes on in the home of the prophet (pbuh) and that they know of his teachings and thought, or chances of the hypocrites spreading false things about the prophet (pbuh). Allah (swt), who is aware of the unseen and what people hide inside themselves, prevented such conspiracies from doing damage through this command and prevented any of the above from taking place, and in order to reinforce the command, he gave the wives the title of “the mothers of the believers” so that everyone would know that marrying them is like marrying one’s own mother! It is because of these reasons that the wives of the prophet (pbuh) must be happy with this deprivation and embrace it.
Throughout one’s lifetime, important issues come up that call for sacrifice, and giving up clear rights; sometimes things like this are needed for certain accomplishments. Clearly, the wives of the prophet (pbuh) gained great honor through marrying him, obtaining such honor calls for such sacrifice (not being able to get married after him). It is because of this that after him, they were greatly respected amongst the Muslim nation, and were happy with their status, and considered such hardships incomparable to the honor they had gained.
This danger becomes even more evident when we see that some had prepared themselves for such a thing (marriage with the prophet’s (pbuh) wives); some verbalizing it, while others having the intention of doing so. One of those whom Sunni commentators have named in this regards is Talhah.
According to Shahid Motahhari, the main reason for it being impermissible to marry the wives of the prophet (pbuh) after him is so that the next husband won't take advantage of the wife’s fame and respect (they were uniquely respected among the Muslims) and to prevent them from becoming a means of reaching political and social aims for selfish individuals.
2- Out of respect for the prophet (pbuh) himself, as can be deduced from the reason why the aforementioned verse was revealed; a number of his enemies were planning to marry his wives after him out of revenge and degradation of the prophet (pbuh).
3- Out of respect for the wives themselves; since they were his spouses and associated with him and bore a great respect amongst the Muslims. Clearly, they weren’t ordinary women, the Quran also corroborates this claim.
 For information on why they have been given such a name, refer to Question 433 (website: 462).
 Naser Makarem Shirazi, Tafsir Nemouneh, vol. 17, pg. 404.
 Morteza Motahhari, Majmou’eye Asar, vol. 19, pg. 431. Also, for further information, see: Tafsir Al-Mizan, vol. 16, verses 1-54 of surah Ahzab; Tafsir Nemouneh, vol. 17, same verses, pp. 405-406. (With help from Question 433 (website: 462).