'Arms' here derives from the use of the word for the tools—the weapons and armor—of an armed man. This term was extended to the design a warrior painted on his shield and the crest he displayed on his helmet so he could be identified by his peers and followers in battle; arms in this sense thus came to proclaim a man's status as a member of the dominant military class; arms soon became hereditary and thus declared not only his personal identity but his lineage.
On ceremonial occasions an 'armiger'—a man entitled to bear hereditary arms—wore over his ordinary dress a garment which displayed his arms; this 'coat of arms' eventually became the term for any full display of arms, including not only the designs on the shield and crest but also ancillary symbols of his family and rank, such as 'supporters' (the falcons in your image) and a slogan.
ADDED:
For those who enjoy the ancient craft, here's my own blazon of the arms of Ghana depicted in OP's illustration. I take the cross to be the principal charge on an unvaried field, but there's room for argument there.
Azure, on a cross vert fimbriated or, a lion passant gardant or; in dexter chief, an asante sword and an oykeame staff or in saltire; in sinister chief, on a rock or emerging from a ocean barry wavy argent and azure, Osu Castle argent windowed and gated gules; in dexter base, on a hillock vert a cacao tree proper; in sinister base, on a hillock vert a gold mine proper. For a crest, on a wreath gules vert and or, a mullet of five points sable fimbriated or. For supporters, two tawny eagles or rousant with wings addorsed; about the neck of each, on a riband gules vert and or, a mullet of five points sable fimbriated or. For a motto, gules on a scroll or, Freedom and Justice.
4I'm not really an expert, but coat means also "something that covers" (shields?), and arms means also weapons (I'm pretty sure that's the case). "it depict weapons and covers something else" – drM. – 2016-03-03T14:46:45.193
1
It makes more sense if you consider definition 2 of arms in ODO.
– Lucky – 2016-03-03T14:52:49.5731
Per the Help Center page concerning what is on-topic here, questions about etymology are better suited to our sister site for English Language & Usage. I have asked a moderator to have a look.
– choster – 2016-03-03T14:52:49.9873@choster Etymological questions per se yes, but this one would have to be edited if the mods migrate it, or I'm afraid the ELU community will frown upon the single definition of 'arms' as plural of 'arm' and the OP not considering other homonyms. – Lucky – 2016-03-03T15:00:42.840
1@Lucky -- huh - that definition kinda makes you think of any company's branding (insignia) applied to stationery, padfolios, coffee cups, etc (essentially the weapons and shields for modern corporate warriors) – Code Jockey – 2016-03-03T19:35:44.623
3I'm voting to close this question as off-topic because, as noted, it's about etymology of a somewhat archaic phrase. – Nathan Tuggy – 2016-03-03T21:14:21.660
2@NathanTuggy: eh? "Coat of arms" is still a going concern for many people, there's nothing archaic about it. – Martha – 2016-03-03T22:07:10.890
4I also don't think this is an etymology question, really: it's a learner encountering an unknown phrase and trying to parse it literally, with amusing but predictable results. My boss gets hung up on idioms the same way: he tries to interpret them literally, is told the (non-literal) actual meaning, and then goes off on a rant about the nonsensicalness of English. – Martha – 2016-03-03T22:09:47.513
@Martha: It's clear that they already understand the meaning; they're asking about the origin. And that's not on-topic here. – Nathan Tuggy – 2016-03-03T22:17:18.183
I'm voting to close this question as off-topic because etymology is specifically stated as off-topic in the help center
– ColleenV – 2016-03-04T00:04:10.3202This discussion is all very interesting, and the etymology aspects would be better off on ELU, but, and I don't want to insult anybody, but if the question arose from the OP simply not realising that both "coat" and "arm" can mean different things than the definitions mentioned, I don't think it belongs on ELU. – Mr Lister – 2016-03-04T08:50:49.520
@MrLister No offence taken! – Denis Kulagin – 2016-03-04T09:09:50.567