How does a meandering river form?
Meanders amplify themselves--the stream tends to erode on the outside of the meander (forming a cut bank) and deposit sediment on the inside (forming a point bar).  Eventually, the meander will propagate so far from the stream's mean path that the stream cuts across it.  Eventually, the old meander will turn into an oxbow lake and fill in.

As to why it erodes on the outside and deposits on the inside, the simple answer is that the water has to travel farther on the outside, so it flows faster and erodes more, while the water on the inside moves slower and deposits its sediment.  A more complete answer involves vortices crossing the river bed carrying sediment from the outside to the inside.