Why neutral wire carry no current?

Usually yes, sometimes no. For instance:

The neutral in a single-phase, 120V (in the US) branch circuit, such as one feeding receptacles, does.

The neutral in a 120/240V circuit feeding a 240V appliance does not.

The neutral in a 480Y feeder feeding a balanced load does not.

A neutral is there because of the possibility that current flow could occur. For instance, in a US household, with 120/240V service, if you plugged in 5 100 watt lamps on one side of the line, and another 5 100 watt lamps on the other hot leg, there would be no neutral current in the service cable feeding the house. The loads are said to be 'balanced'. The 500 watts of power flowing into the first hot leg goes through the first set of lamps, then the second set, then out the other hot wire. Neutral current still flows in the individual branch circuits, of course.

Now, if you moved one of the lamps to the other side, 600 watts would be coming into that side, but only 400 would be going back out the other hot wire, so 200 watts would flow through the neutral.