Translation:I am self conscious.
Chomhfhiosach looks like a compound word in itself (because it ignores the CLC;LLL rule), so chomh = [chou/chuhv], fhiosach = [i-səch]. [Chou-i-səch / Chuhv-i-səch]. Someone who has more Irish, please feel free to correct me. And I didn't use IPA intentionally; no need to complicate things for those who don't know how to read it.
The word comhfhiosach means “conscious” in its philosophical sense, i.e. “aware of one’s own existence or awareness”, so féin-chomhfhiosach really means “self-conscious” in its “aware of oneself as an individual being” meaning, which is why the meaning of féin-chomhfhiosach can be “deliberate, affected”, e.g. in describing someone’s acting performance. It doesn’t mean “self-conscious” in its “uncomfortable with one’s own behavior or appearance” meaning; that English meaning is only approximated with Irish words such as cotúil or náireach, which have meanings closer to “bashful” or “diffident”.
Duolingo teaches "cúthail" for "shy", and "féin-chomhfhiosach" for the "deliberate, affected" meaning of "self-conscious", exactly as that focloir.ie entry you refer to explains.
neamh is also a negating prefix - there are a few exercises on Duolingo using neamhspleách - "independent" (spleách is "dependent").
Other examples are tabhair neamhaird ar for "ignore" versus tabhair aird ar for "pay attention to"
neamhréasúnach - "Irrational", "Unreasonable"
neamhrialta - "Irregular"