Hi there, Elsa!
The Holy Trinity is something that can be quite difficult to get your head round. It took a long time for theologists to actually recognise the concept.
Basically, there is one God. But God has three 'components': Father, Son and Holy Spirit.
If you like, the world has been through three stages. In the first stage, the Father created the Earth and guided His people directly. In the second stage, He sent His son. In the third stage (which we are in now), He sent His Holy Spirit.
Now, Jesus was born of the virgin Mary and was incarnate by the Holy Ghost. He became a man. You are correct that Jesus does not say, "I am God". However, he does say, when the disciples ask him to show them the Father, "Whoever has seen me has seen the Father" because Jesus is one with the Father.
It is Man's nature to commit sin. None of us are perfect. However, as once asked in a song, "What if God was one of us?" This question was actually answered when God the Father sent His son Jesus Christ into the world. Is it possible for a normal human being to be free from sin? No. But if God was a man, would He be able to be free from sin? Yes!
Also, in order for Jesus to be a perfect sacrifice and be sufficient to cover the sins of the whole world, he would have to be God. These things have to be done on the same level, but there is nobody on the same level as God!
As for why Jesus never said that he was God directly, we must remember one thing about Jesus. He is very humble. In the Bible, he refers to himself not as the Son of God, but as the Son of Man!
As you can see from the beginning of John's gospel, Jesus was with the Father right from the beginning.