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Je savais que je gagnais à être connue.
My guess: « Les gens en sont venus à m’aimer de plus en plus à mesure qu'ils me connaissaient et appréciaient mes qualités. »
If I’m on the right track here... What I find interesting and puzzling is that it is « je » who « gagne à être connue » rather than « les gens / on » being the ones who « gagnent à me connaître ».
Je savais qu’on gagnais à me connaître (mieux).
The structure « gagner à » can be applied with a lot of words and expressions. « gagner à [quelque chose]» means to benefit from something. You can for instance say
Le vin gagne à vieillir
Which means that the wine will get better by getting older
As a result, when someone says « je gagne à être connu », it means that he will benefit from being better known. Someone who is making a bad impression at first glance, despite being a valuable person, is someone who « gagne à être connu ». He will improve upon acquaintance.
(I made an edit, read it first) I don't think it's idiomatic, at least not the whole expression.
"Être connu" means "to be famous/well known" , and "gagner à [qq chose]" means "to benefit from [something]".
You can say "gagner à" with anything (same for "être connu"), even though it's a little formal.
You can't say "Je savais bien qu’on gagnais à me connaître". "Je suis connu" and "*on me connaît" are not equivalent, when "connu" means "famous", it's used as an adjective. For the same reason, "je suis perdu" and "on me perd" do not mean the same thing at all.
(btw I'd like to have context for this sentence if you have it, I can't really translate right know. Are you talking about selling a book or being known by your colleagues ?)
EDIT : According to these sources from the Wiktionary (Gagner (see 17), gagner à être connu) the expression seems to 1. be idiomatic and specific to the pair "gagner à" and "connu" 2. means "to deserve to be better known".
However, note that:
Still, I've only seen it used for places, like restaurants, independant movie theaters, or for an author, a novel, a series, etc. I wouldn't use it for people, in a way unrelated to fame.
The part about the inversion still holds, I wouldn't say "Je savais qu’on gagnais à me connaître (mieux)." (at least, not meaning "deserve")
Gagner = Rejoindre la 'position' de ...
position sociale, position géographique ...
Here the verb "gagner à être" is meaning "to have talent in suffisance to pretend behing a famous person". This expression is to the appreciation of anyone who formulates it.
