It could mean that in the town named Dixie, the same things keep happening over and over. Maybe it's a small town where the routine, the events, or the way of life doesn't change much.
The phrase might refer to the fact that Dixie town has a reputation for certain recurring situations. For example, it could be that every year during a particular season, the same events unfold. Maybe there's a local festival that always has the same glitches or a local business that has the same management problems year in and year out. This gives the sense of 'same old story' in Dixie town.
Well, 'Dixie town' might be a specific place. 'Same old story' implies a lack of novelty. It could be about the people in Dixie town always having the same problems, like economic issues, relationship dramas, or social patterns that are stuck in a loop.