The image basically contains (on top of the TeamCity agent itself of course) all the required dependencies. These dependencies are built in exactly the same way (and in the same context) as the ones used for running the code itself (i.e. after building/testing). This basically ensures that the code consistently runs in the same conditions.
Using Docker for this also means I can roll out the agents rather painlessly across a variety of machines. For instance, we have some pretty serious hardware used for HPC and similar stuff, so whenever possible I like running builds on that, since it takes a fraction of the time it would on the more common hardware. Sometimes however those machines aren't available for a number of different reasons, so I can very quickly move my build agents to another machine, or better even, multiple machines.
Using Docker for this also means I can roll out the agents rather painlessly across a variety of machines. For instance, we have some pretty serious hardware used for HPC and similar stuff, so whenever possible I like running builds on that, since it takes a fraction of the time it would on the more common hardware. Sometimes however those machines aren't available for a number of different reasons, so I can very quickly move my build agents to another machine, or better even, multiple machines.