Why were the gospels written down?

Those who were eyewitnesses, or personally knew them, wanted to record the important events and details of the Christian message. They also wanted to assist those who were involved in spreading the Gospel message, that others might also believe, as well as to give an account of the details of Jesus' life and teaching to assist new converts to have a better understanding of the faith. The verses below explain this.

1Forasmuch as many have taken in hand to set forth in order a declaration of those things which are most surely believed among us, 2Even as they delivered them unto us, which from the beginning were eyewitnesses, and ministers of the Word; 3It seemed good to me also, having had perfect understanding of all things from the very first, to write unto thee in order, most Excellent Theophilus, 4That thou mightest know the certainty of those things, wherein thou hast been instructed.

31But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name. There's no proofs that the Evangelists wrote the Books attributed to their names.

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