Electrostatic forces (from slight charge imbalances) help those electrons go around those corners. And for the electrons jumping onto the moving rod from the top, that is what gets them up to speed $v$ rightwards (and back to regular at the bottom right corner). If you imagine an electron that just hopped onto the top of the right wire the protons there are already moving rightwards so there is a positive charge imbalance to the right so the electron is pushed right, over on the left side there is a negative charge imbalance, so the protons are pulled left. They will quickly get a rightwards component of velocity, so for now, let's ignore that and just concentrate on the moving rod.
There are two basic contributors to the magnetic force, one is the motion to the right, that gives a magnetic force up. The other is the current up, that makes a magnetic force to the left. But the magnetic force to the left is just like that cornering force, because of it there is a charge imbalance (negative on the left, positive on the right) enough to make an electrostatic force that cancels it, cancels it enough to keep the charge flowing along the wire.
The total magnetic force is the vector sum of the two (that due to the current, and that due to the rightwards motion v), it is orthogonal to the actual average velocity, which is exactly that needed to get from the upper right corner at $t_1$ all the way to the lower right corner at $t_2$ (which is $L$ units below, and $v(t_2-t_1)$ units to the right of where the upper right corner was back at $t=t_1$. So the velocity is in the direction that the electrons actually move. And the magnetic force is orthogonal to that velocity, so it isn't doing the work.
I'm not sure why your book says that the left side is getting a positive charge (as opposed to the right side), it's the conduction electrons that are moving about. The protons and bound electrons are locked together in a solid lattice so simply get strained by the electromagnetic forces, though there is a net force on the lattice equal and opposite to the net force the lattice exerts on the conduction electrons, and this is what slows down the moving rod, it's pushing those electrons right that are getting pushed left by the current based magnetic force so it will slow down unless something else (like a person) grabs it to push it rightwards.