Breitling is a Swiss watchmaker based in Grenchen, Switzerland. Founded in 1884 by Léon Breitling in 1884 in Saint-Imier. Breitling is known for precision-made chronometers useful to aviators.
The "B" with wings is a called a trademark. That means that it is a stylistic or artistic symbol that is officially registered by the company or brand with a special government agency. For example the Registration Number for the design that you asked about is number 2352162. Because of the registration, no one else in the world can use that symbol in order to sell the same kinds of products as Breitling. Specifically, Breitling told the official government agency that it wanted to use the trademark for: Chorological instruments and chronometrical instruments, namely, watches, wrist-watches, straps for wrist-watches and watchcases, travel clocks, clocks, chronographs, [and] chronometers [or watches].
The stylized or artistic "B" appears to have first been used in the 1940's but it did not have either the wings or the "anchor". What you see are not arrows but the points on the end of a ship's anchor.
The "B" most certainly stands for "Breitling" which was the name of the man who started the company.
Breitling made a lot of watches and was interested in aviation and flying which was a new and growing science in the early 1900's.
Around 1952, Breitling made a watch which they called the "Navitimer"; Probably this was the combination of two words: "navigation" and "timer". Like many Breitling watches made today, it had rings around the glass with a series of numbers. This ring was called a "Slide Rule." A slide rule is an instrument that people used to do math before they had calculators. With the slide rule, airplane pilots could do all sorts of important math using their watch. Today, a pilot would use a computer built into the airplane.
By the time this Navitimer watch for pilots was made in the 1950's Breitling added the wings to their trademark. Since the "Navitimer" and its slide rule were designed for use by airplane pilots, the wings seem an appropriate design to use for a trademark because Breitling was trying to show people that they made watches for aviators and pilots.
By about the 1980's, the Breitling trademark added a ship's anchor. According to Classic Watch In the firm's early days, watches were usually marked on the underside of the dial, so that from the outside it was not easy to see that the watch was made by Breitling. The signature at that time was Montbrillant. [Montbrillant is the name of a street in the City of Geneva, Switzerland where Breitling's offices were located] Around 1930-1932, the watches were signed with the "Breitling" script, which remained until the sixties. The Navitimer was an exception. Beginning in 1952, the AOPA [an organization of airplane pilots] emblem - the large wing with the letters on it, the big swallow - was used, since this watch was intended for pilots. Wily Breitling changed the "Breitling" signature in the sixties, since he was of the opinion that one could not easily read the name of Breitling in script form. He left the typical "B" initial on the dial, and the name could now be read in capital letters. Again it was only the Navitimer that had the two stylized airplanes on the dial as the firm's symbol. The firm's emblems changed more often in advertisements. The present Breitling firm has maintained the old tradition, and the present-day symbol consists of the "anchor" with the letter "B" in the centre, plus the wings, to represent Breitling in the water, on land and in the air.