Patty these are the key sentences from the lesson:

des autres is the contraction here of [de + les autres] (about/of the others), so it implies that we're talking about specific things here (the others), 

whereas using d'autres here is the contraction of [de + autres] (about/of others) is less specific ([some] others).

So putting the examples in some context:

1. You're in the pub and the old man in the corner is telling a bunch of stories about his life in the navy. He spoke of  France and of other (unspecified) countries he had visited.

Il a parlé de la France et d'autres pays (de + autres).

2. "I want to explore France, Germany and Spain because my father has travelled extensitvely. He's told me many stories of (about) France and (about the) of the other (specified i.e., Germany and Spain) countries".

Il a parlé de la France et des autres pays (de + les + autres -> des autres).

Hope it helps!