I'm under the impression you're not asking why a meandering river will keep meandering, but why even a straight river will start to meander in the first place. The first is indeed by erosion of the outside of the bends, and deposition on the inside of the bend, which is fairly easy to understand.
On the origin of meanders, this is a more complicated matter. Theoretically it might be possible to create a perfectly straight and homogeneous canal, with perfectly laminar flow, and in that case you'd expect equal erosion on both sides of the river. However, nature isn't as straightforward (if you pardon the pun), and rivers never start of perfectly straight, neither is the material ever perfectly homogeneous, and natural laminar waters are very rare.
Therefore, even if you would start with a straight canal, some parts of the walls are preferentially eroded over others, causing the canal to no longer be straight. This imperfection will then cause turbulence near the edges of the imperfection, leading to further erosion, thereby creating the seed of a meander.
Also, even if the walls were perfectly homogeneous, the slightest disturbance in the flow will cause the flow to oscillate leading to inhomogeneous forces on the walls and thus uneven erosion.