"a busy rumour" [Rowe], 1620s (earlier "a fancy," c. 1600), figurative use from buzz (v.). Literal sense of "a humming sound" is from 1640s. A "buzz" was the characteristic sound of an airplane in early 20c.; hence verbal sense "to fly swiftly," by 1928; by 1940 especially in military use, "to fly low over a surface as a warning signal" (for example that target practice is about to begin):

The patrol aircraft shall employ the method of warning known as "buzzing" which consists of low flight by the airplane and repeated opening and closing of the throttle. [1941 Supplement to the Code of Federal Regulations of the United States of America," Chap. II, Corps of Engineers, War Department, p. 3434, etc. ]

Meaning "pleasant sense of intoxication" first recorded 1935. The children's game of counting off with 7 or multiples of it replaced by buzz is attested from 1864 and is mentioned in "Little Women" (1868). To give (someone) a buzz (by 1922) is from the buzz that announced a call on old telephone systems (1913). Buzz bomb "V1 rocket" is from 1944.

Old English be- (unstressed) or bi (stressed) "near, in, by, during, about," from Proto-Germanic *bi "around, about," in compounds often merely intensive (source also of Old Saxon and Old Frisian bi "by, near," Middle Dutch bie, Dutch bij, German bei "by, at, near," Gothic bi "about"), from PIE *bhi, reduced form of root *ambhi- "around."

As an adverb by c. 1300, "near, close at hand." OED (2nd ed. print) has 38 distinct definitions of it as a preposition. Originally an adverbial particle of place, which sense survives in place names (Whitby, Grimsby, etc., also compare rudesby). Elliptical use for "secondary course" was in Old English (opposed to main, as in byway, also compare by-blow "illegitimate child," 1590s, Middle English loteby "a concubine," from obsolete lote "to lurk, lie hidden"). This also is the sense of the second by in the phrase by the by (1610s). By the way literally means "along the way" (c. 1200), hence "in passing by," used figuratively to introduce a tangential observation ("incidentally") by 1540s.

To swear by something or someone is in Old English, perhaps originally "in the presence of." Phrase by and by (early 14c.) originally meant "one by one," with by apparently denoting succession; modern sense of "before long" is from 1520s. By and large "in all its length and breadth" (1660s) originally was nautical, "sailing to the wind and off it," hence "in one direction then another;" from nautical expression large wind, one that crosses the ship's line in a favorable direction.