On the meaning of witr and the amount of raka'a's and how it should be performed
First of all witr in Arabic means odd, which reefers to the amount of raka'a's of this prayer.
You basically could pray witr in:
one raka'a, three raka'a, five raka'a all this can be covered and evidenced by ahadith (there are even ahadith for more raka'a's, but they might need some further explanations, research and detail which were not asked as you wanted to know about the three raka'a witr). Witr can be performed in up to eleven raka'a according the shafi'i's and hanbalis. The maliki's only consider a single raka'a is witr and any set of two raka'a's before either as salat al-Lail or shaf'a (synonym for even). On the whole anything between one raka'a at minimum and eleven at maximum can be considered as witr. Note that especially the hanbalis have special details when it comes to how to pray each of these amount (for example 11 raka'a as 5 sets of two raka's and 1 single raka'a, while 9 raka'a with one tasleem but 1 sitting for greeting -tahiyyat- at the end of the 8th raka'a ... for details on the madhhab views read here in the Islamic Jurisprudence according the 4 madhhabs p.439 ff).
The majority of scholars (AFAIK) consider praying at the end of the night prayer when sufficient time is left two raka'a shaf'a (some consider it part of the witr) followed one raka'a witr and in worst case one single raka'a witr (if no time is left).
Among those who consider praying it as a set of three raka'a with one single tasleem and without a sitting for greeting -tahiyyat- at the end of the 2nd raka'a we find the hanafi school of fiqh. This is based on a hadith al-Bayhaqi narrated that says that one shouldn't perform witr like maghrib. See also islamqa fatwa #38230.
Evidences for both ways
- Some evidences for the view held by the hanafi's in first place:
"The night prayer of Allah's Messenger was thirteen Rak'ah, five of which were his Witr, not sitting in any of them except at the end of them. When the Mu'adh-dhin called the Adhan he would stand to perform two light (Rak'ah)." (Jami' at-Tirmidhi)
Imam at-Trimidhi here added a comment that the sahaba and others used to pray witr in five raka'a without a sitting until the last raka'a. Abu Mus'ab al-Madini (Az-Zuhri) أبو مصعب الزهري الْمَدِينِيَّ (a student of imam Malik and narrator of his al-Muwatta' and teacher of scholars such as al-Bukhari, Muslim, abu Dawod, ibn Majah and at-Trimidhi, an Nasa-i heard from him via an intermediate) when asked by at-Tirmidhi on how would they pray witr in nine or seven raka'a answered: they would pray it in sets of two raka'a with tasleem and complete it with a single raka'a.
Imam al Hakim in his al-Mustadrak compiled the hadith (My own translation take it carefully):
The Messenger of Allah Allahs prayers and blessings be upon him used not to say the tasleem in the (first) two rakahs of Witr.
Aishah told him: "The Messenger of Allah (ﷺ) would not say the taslim for two rak'ahs during witr." (Sunan an-Nasa-i)
This means he prayed with one tashahud (tahiyyat) and one tasleem
The Messenger of Allah (ﷺ) used to pray thirteen rak'ahs, observing six rak'ahs in pairs including the two rak'ahs of dawn prayer. He would observe witr and five rak'ahs. He sat only in the last of them. (Sunan abi-Dawod)
Indirectly shows that the practice of one single tashahud is acceptable.
- Now to the evidences for those who consider praying witr as two raka'a + one:
Witr is a rak'ah at the end of the prayer. (Sahih Muslim, sunan an-Nasa-i)
I asked Ibn 'Abbas about the Witr prayer. He said: I heard the Messenger of Allah (ﷺ) as saying: It is a rak'ah at the end of the night prayer. (Sahih Muslim)
A man who lived in the desert asked the Messenger of Allah (ﷺ) about the prayer at night. He made a sign with his two fingers-in this way in pairs. The witr consists of one rak'ah towards the end in night. (Sahih al-Bukhari, Sunan abi Daowd)
From the practice of the sahaba:
Nafi' told that 'Abdullah bin 'Umar used to say Taslim between (the first) two Rak'at and (the third) odd one in the Witr prayer, when he wanted to attend to a certain matter (during that interval between the Rak`at). (Sahih al-Bukhari)
So basically one could conclude that both ways have been practiced by the Prophet () and are correct (see also islamqa #46544).