The first hint about cause is that the problem shows up when there is pressure on the drive train, either from being on the road or applying the rear brakes. As the tension on the lower section of chain between chainring and pulley is not affected by pressure on the drive train the problem must be occurring before that point. That means the interaction between chain and chainwheel, such that the chain is unable to disengage when it has been solidly seated in the chainwheel teeth.

What you are experiencing is chain suck, and the most likely cause is that your chain is new or not worn and the small chainwheel is sufficiently worn or damaged - or a combination. Although not a "shark's fin" the small chainring is definitely more worn than the large. I'm also wondering if that chainring was often ridden in more of a cross-chain combo, which could have produced some burrs. You could certainly try reversing the chainring, as well as making sure the sides of the teeth are smooth. A new chainring would certainly solve the problem. Here's quite a treatise on chain suck: http://www.fagan.co.za/Bikes/Csuck/

This needs to be corrected as soon as possible, as it can get suddenly worse, especially on a climb, resulting in the chain wrapping around, causing the pedals to seize and possibly a fall or damaged parts.