I'm reading a motivational book. The author often writes "Inspire on!", it's a kind of slogan. What I don't really understand is if it means more: "inspire other people" or: "be inspired when you're doing something", or both.
It depends on the context of the book, but I can see two possible meanings for the phrase.
Continue to feel inspired, or
Inspire [others] on!
Without knowing more context, it is difficult to tell if the author is telling the reader to feel inspired, or to inspire others. However, since it is a motivational book, it is likely the first meaning.
I hear some people nowadays use "inspire" as a reflexive verb -- e.g. "I inspired myself with a nature walk" -- so perhaps the author did intend to use it in that sense.
However, whether it was meant that way or in a "inspire others on" sense, the slogan seems very awkward (perhaps to the point of being ungrammatical), because "inspire" is a transitive verb. In just about any other example I can think of, when a verb is paired with "on" that way (e.g. "Shine on, harvest moon" from early 1900s song; "Roll on, thou deep and dark blue Ocean—roll!" from Lord Byron's Childe Harold, Canto IV, stanza CLXXIX), the verb is an intransitive one.
Just because someone writes something in a book does not make it good English. As has been pointed out, "inspire" is a transitive verb and requires an object. I suspect the author meant "aspire."