How was the distance 7 feet, 9 and a quarter inches, established from the dartboard to the oche?

The standardisation of the throwing distance took place around the same time, although there is still more than one "standard" in use. It is said that the throwing distance was marked by placing three crates end to end from a brewery called Hockey & Sons (which supplied beer to the Southwest of England). The crates were three feet long, making the distance from the line to the board nine feet. The size of the Hockey & Sons crates was eventually reduced to two feet, and four crates lined up to mark the distance (eight feet). The 8-foot distance remained the standard for many years -- and still exists in some places.

http://www.darts501.com/History.htm

Employees at the brewery firm of Hockey & Sons are credited with establishing the standard throwing distance (there are actually two standards - more on that in a moment). It seems that the brewers placed three of their beer crates end to end, drew a line, and threw from there.

Since each crate was 3 feet long, the throwing line was set at 9 feet from the board. Later, the standard Hockey & Sons beer crate was shortened to 2 feet, so 4 crates were lined up, which set the line at 8 feet. This 8 feet standard is still in use in many places, though local variations exist. The general international standard is 7 feet 9 and 1/4 inches. The throwing line was called the "hockey" after the company, though later the name was shortened to "oche" (pronounced "ockee").

http://www.britainexpress.com/History/pastimes/darts.htm

According to Patrick Chaplin this theory is not true.
http://www.pdcworldchampionship.co.uk

I can't locate that reference page, here is Patrick Chaplin's website.
http://www.patrickchaplin.com/

and the related history of darts page. I imagine the information is here, somewhere.
http://www.patrickchaplin.com/History.htm