Do species ever move into new territory and then die out because of the local diseases?
So how come the story is always that the invading species kills off
  the locals, when you would expect the invading species to have
  hundreds of totally new threats and strange foods to handle, would be
  unable to cope, and would die out?


There is a problem of observation bias for these two scenarios.  In the first scenario a few members of an invasive species enter a new range, transmit a novel disease to the pre-existing species, which suffers a massive (easily observable) die off. In the second scenario, a few members of an invasive species enter a new range, catch a local disease, and die in short order (unobserved and unnoticed). 

In other words, a few individual deaths from any cause will generally go unnoticed. Thousands or millions of deaths of deaths from a novel disease are more likely to attract attention.