What does "on-line" mean, as used in man(1)?

In contrast to a printed (hard-copy) manual, which you could read off-line (while not using a computer).

The term dates back (at least) to time-sharing systems. Users may have had a terminal which could be used for typing text, punching paper tapes. But they were only able to use the computer when they were on-line (the "line" referring to the communications link from the terminal to the computer).

Lots of English is that way: you likely use terms which on reflection you might not consider up-to-date.

The word "on-line" is used in the sense "operating under the direct control of, or connected to, a main computer." Reading a manual "on-line" is therefore the same as reading it "on the computer".

This is in contrast to "off-line" in the sense "operating independently of, or disconnected from, an associated computer." Reading a manual "off-line" is therefore the same as reading it by some other means, probably on paper (or asking the in-house wizard who knows every manual by heart).