Short Answer

The Prophet allowed this because they loved him and because their dedication to him would serve the purpose of making his enemies know how strong and dedicated the Muslim community was. Some Companions continued the practice of honoring his relics and seeking blessings from them, but there are no more authentically attributed things from him remaining today.

What the Sunnah Says

Asma bint Abi Bakr…said: “Here is the cloak of the Messenger of Allah.” She brought out…a cloak made of Persian cloth with a hem of brocade, and its sleeves bordered with brocade and said: “This was Allah’s Messenger’s cloak with Aishah until she died. When she passed away, I got possession of it. The Messenger of Allah used to wear it, and we washed it for the sick so that they could seek cure thereby.” (Sahih Muslim #2069a)

“The daughter of the Messenger of Allah (Zaynab) passed away and he said to us: “Wash her thrice or five times or more, if you see it necessary, and when you finish, notify me.” So when we finished, we informed him and he gave us his waist-sheet and told us to shroud her in it.” (Al-Bukhari #1191)

Mahmud ibn Ar-Rabi said: “Whenever the Messenger of Allah performed ablution, the Companions almost fought over the excess water.” (Al-Bukhari #186)

Abu Juhayfa narrates that the Messenger of Allah came to us at noon, and water for wudu (ablution) was brought to him and he performed wudu. Hence, people began to take the remaining water left from it and wipe their bodies with it (for blessings)…Abu Musa said: “The Messenger of Allah asked for a container of water and washed both his hands and face in it and then spat a mouthful of water in the tumbler and said to both of us (Abu Musa and Bilal): “Drink from the container and pour some of its water on your faces and chests.” (Al-Bukhari #185)

Anas said that Umm Sulaym used to put out a mat of leather for the Prophet and he would take a nap in her house. When the Prophet fell asleep, she took some of his sweat and hair and kept it in a bottle, then she put it in some musk perfume. He [Thumaamah ibn ‘Abd-Allaah ibn Anas] said: When Anas ibn Malik was dying, he left instructions that some of the perfume should be put in his burial perfume. (Al-Bukhari #6281)

Anas ibn Malik narrates that the Messenger of Allah came to Mina. He went to the Jamrah and threw stones at it. Thereafter he went to his tent in Mina and sacrificed the animal. He then said to the barber, turning his right side and then the left side to him, “shave the hair”. He then began to give these hairs to the people. (Muslim #1305)

Uthman ibn Abdillah ibn Mawhab said: “My family sent me to Umm Salama with a cup of water…. (and Umm Salama brought out) a silver bottle which contained one of the hairs of the Messenger of Allah. And it used to be such that if anyone came under the evil eye or anything else, they would send her their utensil. I looked into the silver bottle and saw some reddish hairs.” (Al-Bukhari #5896)

What the Scholars Said

Imam Al-`Ayni says: “Umm Salama had some of the hairs of the Messenger of Allah kept in a silver bottle. When people became ill they would go and obtain blessings from these hairs and they would be healed by means of its blessings. They would take the hair of the Messenger of Allah and place it in a cup of water and drink the water in which the hair was placed and by this they would be healed.” (Umdat al-Qari sharh Sahih al-Bukhari, 15:94)

Imam an-Nawawi says: “This narration (of the cloak with Aishah) is proof for the recommendation of seeking blessings (tabarruk) through the traces of righteous people and their clothes.” (Sharh Sahih Muslim, #1571)

Shaykh Al-Albani said: “Although the Prophet approved of the Companions seeking blessings from his relics and touching them during the campaign of al-Hudaybiyah and on other occasions, that was for an important purpose, especially on that occasion. That purpose was to scare the disbelievers of Quraysh and to highlight the extent of the Muslims’ devotion to their Prophet and their love for him, and their willingness to strive hard in serving him and showing him respect. But what we cannot overlook or conceal is the fact that after that campaign, the Prophet discouraged the Muslims, in a wise and gentle manner, from seeking blessings in this way, and he instructed them to do righteous deeds, which would be better for them before Allah. This is what is indicated by the following narration…the Prophet performed wudu and his Companions began wiping themselves with his leftover water. The Prophet said to them: “What made you do that?’ They said: Love for Allah and His Messenger. The Prophet said: “Whoever wants to love Allah and His Messenger, or wants Allah and His Messenger to love him, let him be truthful when he speaks, and let him fulfil his trust when he is trusted, and let him be kind to his neighbor.” This is a proven hadith with several chains of narration and corroborating evidence in the Mu‘jams of at-Tabaraani and elsewhere.” (Nasir ud-Din Al-Albani, At-Tawassul, 145)

Advice

Firstly, it was not a common practice that the Companions would go to regularly seek blessings from the relics of the Prophet, so it should not be a regular practice, nor should anyone think they are missing out on something important in Islam. Secondly, it is probable that all of the Prophet’s relics were lost over time. For example, when the Tatars attacked Baghdad in 656 AH, it is reported that they burned the cloak (burdah) of the Prophet. When Timurlane attacked Damascus in 803 AH, it is said that shoes which supposedly belonged to the Messenger of Allah disappeared. Thirdly, there is no proof that any of the relics of the Prophet exist now. Some people claim to have hairs or other items from the Messenger of Allah but they have no proof to validate that. Any imposter in history could have made up a story about some relic belonging to the Prophet. The same rigorous system of authenticating a hadith of the Prophet should be applied to any claim of something belonging to the Prophet. I do not know of any that have met this criteria. Professor Ahmad Timur Pasha said, after listing the relics attributed to the Prophet in Istanbul, “It is obvious that some of these relics may be genuine, but we do not see any of the trustworthy scholars stating definitely whether they are genuine or not. Allah knows best about them. But with regard to some of them, we cannot hide our suspicion concerning them…there is no sound evidence to prove that any of the hairs that were in the hands of the people could be attributed to the Prophet because the hairs that the people have were some of that which had been shared out among the Companions, but it is difficult to tell what is genuine from what is not. (Al-Aathaar An-Nabawiyyah, 78-82)