What’s the origin of the phrase “bat(s) in the belfry” and what does it mean?
I believe that it means that a person is slightly loopy/crazy/insane. A belfry is the area where bells are hung in the top of tower, like in a church steeple.
I don’t know where the phrase originated but it’s meaning is quite clear, a person with bats flying around in their head would be less than intelligent 
I offer this for analysis by greater minds than mine. I paraphrase some entries from Cassell’s Dictionary of Slang.
The entry for belfry (1900’s-1910’s) states that it is slang for head (no, not that kind of head).
The entry for bats (20th C.) gives us crazy, insane or eccentric.
However batty (20th C.) is also defined as crazy, insane or eccentric.
Interestingly there was a 19th century barrister called Fitzherbert Batty whose certification as mad in 1839 caused much interest (probably among his clients). That said, it is moot that Mr. Batty is the source since the chronology militates against it
Make of that what you will.
Bats in his belfry seems to have become the phrase about 1899 in the US. It replaced the former phrase wheels in his head. Can’t say why. Probably something used by a humorist or a play, and it was picked up as sounding better.
I suspect “bats” preceded “bats in the belfry”. Bats tend to dart around and rapidly change directions when compared with the flight of the typical bird (although many insects fly in a fashion similar to a bat’s trajectory), & by metaphorical analogy represent the thinking of the lunatic.
Belfries are where you might often find bats, and they are up top of the structures that have 'em, ergo represent the head.
Phrases like this with alliteration or rhyming are likely to enter the vernacular even if they don’t make as much sense as this one (e.g., drunk as a skunk).
“Toys in the attic” means the same thing, though I think is chiefly British, popularized by the Aerosmith tune/album of the same name.
But having bats in the belfry means you’re nutty. Having the wheels turning in your head means you are thinking (presumably sane) productive thoughts.
Just have to share:
There is a town in Montana named Belfry.
The high schools mascot name is the Bats.
(as of mid-90’s that is…assume it still is)
That’s what it means to you, today. Who knows what it meant to someone in the 1880’s?
In Janet Kagan’s very enjoyable SF novel Mirabile, one of her characters makes the point that a church belfry that’s infested with bats is no longer being used, and is almost certainly decrepit. A functioning belfry would have bells, and they would be rung frequntly. This would scare off, and possibly deafen, any bats that might live there.
So if you have “bats in the belfry”, your uppermost part is unused and in serious disrepair. 
This web site and this one give the earliest written uses of the term in the US in the 1890s. Maybe there were enough broken-down churches at that time to give this alliteration currency.
Toys in the attic;
Bars in the window;
Surely gone fishing;
They must have taken my marbles away…