Why do Jehovah's Witnesses believe that angels are doing the talking between God and mankind? Jehovah's Witnesses believe that the conversations between God and the people are carried out by angels, but the angels are not mentioned in the Bible. They are merely assumed to be present, because no one has seen God. How accurate are the angels as messengers? Is it assumed that the angels carry God's word word-for-word or paraphrasing God's words?
This could probably be a lengthy answer, but it would be better to summarize one core belief, which should easily answer the remainder of your question.

Jehovah's Witnesses believe by a number of scriptural proofs that Christ himself is an angel, the highest ranking one, and he is the primary means that through which the speaking between God and man occur. 

Since "angels" are the subject here, there is for instance Jude 1:9 [Amplified Bible]:


  9 But when [even] the archangel Michael, contending with the devil,
  judicially argued (disputed) about the body of Moses, he dared not
  [presume to] bring an abusive condemnation against him, but [simply]
  said, The Lord rebuke you!


This of course is stated on the basis that Jehovah's Witnesses understand that Michael is the same person as "Jesus" (Michael being his pre-earthly angelic name).

Contrast this to how God himself deals with Satan's taunts regarding one of his servants in the first chapters of Job - the reaction is far more authoritative and altogether different. Simply, there is a stark difference between how God (not an angel) and Jesus (an angel) deal with Satan's challenges. 

So if Jesus, the "Firstborn of creation" (Proverbs 8:22) is an angel, and he is also called "The Word", the primary means of which God's word is transmitted to us, then that is probably the first and foremost answer to your question.

Then there is Hebrews 2:2:


  2 For if the message given through angels [the Law spoken by them to
  Moses] was authentic and proved sure, and every violation and
  disobedience received an appropriate (just and adequate) penalty,


Without a complicated run-down of every scripture, Jehovah's Witnesses understand that God's angels are very obedient (indeed perfect) and just like human copyists of scripture, have transmitted messages exactly as directed. Of course Jehovah's Witnesses do not put a huge amount of undue attention on angels themselves, and follow the example angels set in directing glory go God alone. Revelation 19:10:


  10 Then I fell prostrate at his feet to worship (to pay divine honors)
  to him, but he [restrained me] and said, Refrain! [You must not do
  that!] I am [only] another servant with you and your brethren who have
  [accepted and hold] the testimony borne by Jesus. Worship God! For the
  substance (essence) of the truth revealed by Jesus is the spirit of
  all prophecy [the vital breath, the inspiration of all inspired
  preaching and interpretation of the divine will and purpose, including
  both mine and yours].


In a looser sense, the word angel also means "Messenger" and indeed many messengers are involved in the transmitting of God's word - including humans in the spreading of the word throughout the earth.

I have no doubt that there is a more complete answer than this to be made, but this is a general overview. Its most important to state that Angels are not the focus of JW beliefs, and they do not receive undue attention.