The answer to that is, surprising as it may sound to those raised in certain religious backgrounds, it doesn't necessarily save you. After all, the demons believe in Christ and he himself said in Matthew 7: 21-23 that “Not everyone who says to Me, 'Lord, Lord,' shall enter the kingdom of heaven, but he who does the will of My Father in heaven."
But let us assume that the belief in question is the sort that saves. Why does it do so? It cannot simply be a matter of thought. The current world situation has revealed many Christians who think they believe in Christ not to do so really. They have gone along with the worldly agenda without questioning it for the most part. Their leaders certainly have which fact, incidentally, shows the absolute necessity nowadays to think for yourself and not follow leaders, even (especially?) religious leaders. We are called to save ourselves. Through Christ certainly but we must adopt a creative faith meaning one that is alive to the movement of spirit. It cannot just take its nourishment from outside whether that be community, religion, religious teaching, scripture, whatever. It must also grow from the Word of God that is within every human heart and, in fact, this is now the more important of the two factors. Our spiritual vitality must come from within if we are to have any spiritual vitality.
Believing in Christ saves when it is the true inner belief because it is the tuning of the soul to spiritual reality, the opening up of the heart to divine truth, and this then starts the process of spiritual transformation. Christ is truth. Accepting Christ is accepting truth. Not accepting him is rejecting truth. If you reject truth you cannot be saved because you have put yourself outside reality. If you accept truth, really accept it through Christ who is its personification, that truth will grow within you.You will start to become it yourself and this is salvation. You accept truth, you become truth. There is no other salvation.
Jesus said that he who does the will of his Father will be saved so the only belief that counts for anything is that which affects the whole being. It is not that God is pleased if someone does his will and then says "Ok, you're in"! It is that this very belief spiritually changes a person by removing the dead weight of unredeemed matter and converting the soul into a real spiritual thing, one of the same quality as the spiritual world. Only heavenly substance can enter heaven.