this process appears to be a bit buggy, but it is possible to get it to work. Of course, it looks nothing like it did in the trailer for the DLC. There will be an achievement/trophy for starting an arena fight when settlers are watching, so here is what you need to get settlers to watch arena fights.
In order to get settlers to come and watch the fight, you need 2 things. A caged arena and a quiet bell. You can find the quiet bell in the "misc" section of the cages category. The quiet bell looks like a set of traffic lights. When built the lights will be red and when activated they turn green. When turned on the bell is meant to attract settlers to come and watch the fight. Of course, this is the bugged part so you need to build a workaround.
First, build a regular bell and ring it. This will bring all of the settlers from your settlement to the bell. Right next to this have your quiet bell. Ring the quiet bell and it should attract the attention of the settlers (it doesn't work well). I had an entire settlement of 24 people come to me when I rang the town bell. Only 2 of these sat down to watch the arena.
Hopefully this gets patched in the future, but for now, we are stuck with a buggy system.