だもの is more often used by female speakers and it means ですから or だから。
This ending is not necessarily gender specific but it is colloquial word ending and carries something emotional or has certain softness about it: I'm male but I might still use it if I see it fit but that won't be the case usually.
Should not be used in formal occasions or by a male speaker without thorough understanding of the word.
Anyway, this is just my wild guess but the word seem to be related to だもん which is used by children or young ladies to imply that an opinion they are expressing is amenable or to express petulant unhappiness; like pouting lips.
As だもん is used in such a way as the above, feminised form, だもの, emphases maturation in contrast to だもん, as well as carrying petulance that だもん has. Thus, だもの does match well with an imagery of a mature female person who knows how to use her own gender as an advantage.
Therefore, what you pointed "だもの feels like the expression of a reason, or more like with women an excuse to get out of responsibility, doesn't it ??"
is not totally off but there are some other usages as well, so I can't it is not exactly on the mark: the ending is almost literary expression nowadays.