Answer:
First of all, not all questions are fair. Sometime people put statments in the questions which are not true. Therefore, we have to clarify some facts before we go any further:
"Prophet Muhammad, peace be upon him, did not marry Ayesha when she was only 6 years old. Her parents offered marriage as was customary in their culture at the time. However, the prophet, peace be upon him, did not accept this and they waited a number of years before offering marriage to him again."
Next thing is, we have to understand that Islam is all about "Rights and Limits". The Quran clarifies what everyone's rights are and what the limits are as well.
It was through this matrimony of marriage to the prophet, peace be upon him, that we found the example of the meaning of the limitations set forth in the fourth chapter of the Quran, entitled "The Women":
"O you who believe! You are forbiddent to inherit women against their will." - Surah An-Nisaa' (Chapter 4:19)
Women around the world used to have no rights and men knew no limits in dealing with the women. One of the problems was that a man could marry off his daughter at any age and she could not refuse.
Prophet Muhammad, peace be upon him, was offered the marriage to Ayesha by her father and mother on more than one occassion. Once when Ayesha was only six. She was summoned by her mother to come into the house to hear the proposal of her father to prophet Muhammad, peace be upon him, the prophet did not accept this offer, even though it was customary at that time to do so, but as she was not yet of age.
[Note: her father is the one offering the marriage to his lifelong friend (Muhammad, peace be upon him)] The prophet, peace be upon him, did not marry her at the time and she says she returned back to place outside.
NOTE: She did not marry him and returned to her friends and went back outside to play.
[Again - please note: they did not get married at all on this occassion]
A few years later when she was old enough to marry according to Islam (she was able to bear children) she did accept the proposal of marriage and she did marry the prophet, peace be upon him.
We find that the girl must be old enough to have children and to be able to determine what her will is. The prophet, peace be upon him, married her at the youngest age a girl can get married. That is, she was old enough to be considered eligible for marriage and having enough wisdom to make such a choice even at her young age.
Also, she was a virgin. This was to show Muslims about how to treat the young girls when they get married and not to rush into having sex with them until they are ready and fully prepared.
All of this is recorded in the hadeeths narrated by Ayesha herself in regard to the treatment that she received from the prophet, peace be upon him.
As a reward for her commitment to Allah and to Islam, Allah honored her in surah an-Nur by clearing her of any suspicion of illegal sex with a companion, as some liars had charged against her. There were ten verses revealed in regard to this incident alone about Ayesha.
An-Nur [24:11-21]
Verily! Those who brought forth the slander (against Ayesha) are a group among you. consider it not a bad thing for you. Nay, it is good for you. Unto every man among them will be paid that which he had earned of the sin, and as for him among them who had the greater share therein, his will be a great torment.
Why then, did not the belivers, men and women, when you heard it (the slander) think good of their own people and say: "This is an obvious lie?"
Why did they not produce four witnesses? Since they have not produced witnesses; then with Allah they are liars.
Had it not been for the Grace of Allah and His Mercy unto you in this world and in the Hereafter, a great torment would have touched you for that whereof you had spoken.
When you were propogating it with your tongues, and uttering with your mouths that whereof you had no knowledge, you counted it a little thing, while with Allah it was very great.
And why did you not, when you heard it, say: "It is not right for us to speak of this. Glory be to You (O Allah) this is a great lie."?
Allah forbids you from it and warns you not to repeat the like of it forever, if you are believers.