Why do we observe silent prayer with our hands raised to our face?

The Holy Prophet ^sas would sometimes raise his hands for prayer, and we follow that example. For example:

While the Prophet ^sas was delivering the khutbah [sermon] on a Friday, a man stood up and said, “O, Allah’s Apostle! The livestock and the sheep are dying, so pray to Allah for rain.” So he ^sas raised both his hands and invoked Allah (for it).

(Sahih al-Bukhari, Book 11, Hadith 56)

A companion narrates:

I told the Prophet ^sas about our and Abū Amir’s news and how he had said, “Tell him to ask for Allah’s Forgiveness for me.” The Prophet ^sas asked for water, performed ablution and then raised his hands, saying, “O Allah’s Forgive Ubaid, Abu Amir.” At that time I saw the whiteness of the Prophet’s ^sas armpits. 

(Sahih al-Bukhari, Book 64, Hadith 353).

Another companion narrates:

When we reached the Prophet ^sas, we mentioned to him the whole story. On that, the Prophet ^sas raised both his hands and said twice, “O Allah! I am free from what Khalid has done.” 

(Sahih al-Bukhari, Book 64, Hadith 368)