'Browned off' is most widely associated with the British forces, especially the RAF, and is generally dated from around the time of WWII. There's no definitive documentary evidence for that association. Partridge's Dictionary of Slang, 1961 states that it was "Regular Army since ca. 1915" but, as is commonplace with Partridge, no evidence is offered. In Words at War, Words at Peace, Partridge's 1948 collection of essays on military language, he also asserts that 'browned off' "arose in the [British] Regular Army, apparently in India".
Why 'browned'? No one knows. Things may be said to be 'browned off' in a purely literal sense, for example, food that is grilled, crops that are ripening, a person's skin when tanned by the sun. In the 19th century the expression was in use to mean 'ruined; rendered useless'. It appears with that meaning as early as September 1883, in an article about the Canterbury fruit harvest in The Oxford Journal:
Fortunately the weather has been damp and cool... because otherwise the remaining fruit would have been "browned off," and rendered useless.
None of the above brings to mind any feelings of boredom or sadness. It may be that the expression links to two other synonymous phrases, also widely ascribed to the military - 'brassed off' and 'cheesed off'. No one knows why 'brassed' was chosen either. Copper coins, which are of course brown, used to be known as 'brass' and there's a possible, if entirely speculative, link between 'browned' and 'brassed'. That link, even if it exists, doesn't go anywhere to explain the 'sad; bored' meaning.
So, we know very little about 'browned off'. My best guess is that 'browned', 'brassed' and 'cheesed' don't refer to anything specific; they just sound, with 'off' added, evocative of a state of boredom. What we can say is that 'browned off' is known in print from at least 1941. An early example comes from the Somerset newspaper The Western Gazette, March 1941, which reprinted a letter from a soldier serving in the Royal Wiltshire Yeomanry
"We are still miles from any real civilisation and at nights we get a bit browned off."
That reinforces the military origin theory but does nothing to explain the derivation. It is worth noting that 'browned off' probably predates 1941, by a year or two at least. A greyhound of that name was repeatedly reported as a runner in races from July 1940 onward.