Removal of Pubic and Armpit Hair: Why?
Question

What is the reason for shaving pubic hair and armpit hair? What did they use to shave it with at the time of the Prophet (blessings and peace of Allah be upon him)? Is it sufficient just to cut it or must it be shaved?
Praise be to Allah.Is shaving pubic hair sunnah?
The Sunnah indicates that it is prescribed to remove pubic hair and armpit hair. Al-Bukhari (5889) and Muslim (257) narrated from Abu Hurayrah (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “The fitrah is five things – or five things are part of the fitrah – circumcision, shaving the pubes, cutting the nails, plucking the armpit hairs, and trimming the moustache.”
Why do Muslims shave pubic hair?
The wisdom behind the prescription of removing the hair from these two places – and Allah knows best – is that removing it helps one to attain a perfect level of cleanliness and prevents what could emanate from them of bad smells if the hair was left without removing it. And there are other reasons and wisdoms behind it.
Al-Hafiz Ibn Hajar (may Allah have mercy on him) said: 
“These characteristics of the fitrah may serve some religious or worldly interests, that one may notice by reflecting upon the issue, such as: 

improving one’s physical well-being; 
cleansing the body thoroughly; 
taking precautions to ensure purity when doing ghusl or wudu’; 
doing a favour to one’s friends and companions by avoiding unpleasant smells that may offend them; 
differing from the practices of the disbelievers such as the Magians, Jews, Christians and idol-worshippers; 
obeying the command of the Lawgiver; 
preserving that which is mentioned to in the verse in which Allah, may He be exalted, said (interpretation of the meaning), “and [Allah] has given you shape and made your shapes good (looking)” [Ghafir 40:64], because by doing so one is preserving that beautiful image – it is as if the verse implies: I have given you beautiful shapes, so do not distort them with anything that may make them ugly, and take care of them so that they will continue to be beautiful, for taking care of them is a kind of adhering to dignity and maintaining harmony with others, because if a person appears handsome or beautiful, that makes others feel at ease with him, so people will listen to what he says and appreciate what he says, and vice versa.” (Fath al-Bari)

How did the Companions remove pubic hair?
What was well known at the time of the Prophet (blessings and peace of Allah be upon him) was that they would use a razor to shave their hair .
Al-Bukhari (5079) and Muslim (715) narrated that Jabir ibn ‘Abdullah (may Allah be pleased with him) said: We were with the Messenger of Allah (blessings and peace of Allah be upon him) on campaign, and when we approached Madinah, we wanted to enter the city straight away, but the Prophet (blessings and peace of Allah be upon him) said: “Delay it until we enter at night, so that the one who is dishevelled may tidy herself up and the one whose husband is absent may shave her pubic hair.”
Al-Hafiz Ibn Hajar (may Allah have mercy on him) said in Fath al-Bari: This refers to the woman whose husband is absent. What is meant is so that she may remove her pubic hair, and the word used in this hadith refers to shaving, because that is the usual method of removing hair, but that does not mean that it is not allowed to use something other than a razor. End quote.
Al-Bukhari (3989) narrated the story of Khubayb ibn ‘Adiyy (may Allah be pleased with him), in which it says:… when they [the disbelievers who had captured him] decided to kill him, he asked to borrow a razor from one of the daughters of al-Harith so that he could shave his pubic hair, and she lent it to him…
It says in Musnad al-Imam Ahmad (26705), in the hadith of Ma‘mar ibn ‘Abdullah (may Allah be pleased with him): … When the Messenger of Allah (blessings and peace of Allah be upon him) had slaughtered his sacrifice in Mina, he instructed me to shave his head. So I picked up the razor and stood by his head. The Messenger of Allah (blessings and peace of Allah be upon him) looked me in the eye and said to me: “O Ma‘mar, the Messenger of Allah (blessings and peace of Allah be upon him) has let you take hold of his earlobe when you have a razor in your hand…”
How to remove pubic hair
The Sunnah with regard to the removal of pubic hair is to shave it. In the case of armpit hair , the Sunnah is to pluck it, but if a person simply trims it, there is nothing wrong with that, but it is not what is best.
Ibn Qudamah (may Allah be pleased with him) said: 
“Shaving the pubic hair is mustahabb and is part of the fitrah , and it is very inappropriate to leave it, so it is recommended to remove it, and whatever means one uses, there is nothing wrong with it, because the point is to remove it. It was said to Abu ‘Abdullah (i.e., Imam Ahmad): Is it acceptable for a man to cut his pubic hair with scissors, even if he does not cut all of it? He said: I hope it is acceptable, in sha Allah.” (Al-Mughni, 1/65)
An-Nawawi (may Allah have mercy on him) said: 
“Shaving the pubic hair is called istihdad in Arabic because iron (hadid) is used, i.e., the razor. That is Sunnah, and the purpose behind it is to cleanse the area. The best is to shave it, but it is permissible to cut the hair, pluck it or remove it with a depilatory paste…. With regard to plucking the armpit hair, that is Sunnah according to consensus. The best is to pluck it for the one who can stand it, but it may also be done by shaving or using a depilatory paste. It was narrated that Yunus ibn ‘Abd al-A‘la said: I entered upon ash-Shafi'i (may Allah have mercy on him), and the barber was with him, shaving his armpits. Ash-Shafi‘i said: I know that the Sunnah is to pluck it, but I cannot stand the pain.” (Sharh Muslim by an-Nawawi, 3/149)
And Allah knows best.