I know it is a reference of some sort to "space", but in what sense and why?
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The phrase became popular for being in the opening narration of Star Trek:
The idea is that the space is the final frontier for the man to cross to be able to explore new worlds. The previous frontiers would have been on Earth itself, when mankind crossed oceans to explore new continents, for example.
"The frontier" is often used to describe the farthest reaches of civilization (or whatever's viewed as civilized by the speaker). So in the early years of the US, what was considered the frontier by settlers advanced slowly West across the continent. The frontier is usually considered the province of the explorer or the early settler.
By the 60s (when Star Trek popularized the phrase as noted by Bruno R.), humanity had to look beyond the surface of the Earth for the frontier.