And why is it necessary?
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You know how when you go round a corner in a car, you are thrown to the outside of your seat? That is centrifugal force. If you were on a bicycle that was upright then that force would tip you over as you went round the corner, so you need to lean into the corner. By leaning inwards the force of gravity that is trying to tip you towards the centre balances the centrifugal force trying to tip you the other way.
But how do you achieve that initial inwards lean? You might think that you could just lean your body inwards, but that just causes the bike itself to lean outwards. So instead what you do is steer outwards slightly (without noticing). Effectively this moves the wheels outwards while your body carries on in a straight line – the overall result is that you are now leaning. Then you steer inwards to maintain the lean and turn the corner. At the very end of the turn, again without noticing, you steer into the turn extra to pop out of the lean and carry on your way.
You can do all this without your hands on the handlebars because the geometry of the bike (especially the way the steering axis is raked forward) allows you to steer the front wheel by leaning the bike frame.
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