In a 2002 interview with OOR magazine, Boards of Canada explained that "1969" "refers to a specific period in the history of a religious group, and at the same time the period in general, the hopefulness of a forward-thinking generation that wasn't aware of what was coming in their collective future" (OOR Magazine, 2002).
The "religious group" referred to, given the lyrics, is no doubt the Branch Davidians.
It is worth noting that 1969, Sixtyniner and June 9th all reference "69". Also, it is one number below the number 70, (Music70 being Boards of Canada's record label). 70 is also referenced in songs "The Smallest Weird Number" and "Sixtyten".
Samples/Lyrics
"Although not a follower of hseroK divaD" (David Koresh, reversed), she's a devoted Branch Davidian (chorus)
Note that the live version of '1969' heard at the Lighthouse Party does not reverse the name David Koresh
Same words occur in Poppy Seed, BoC remix. The person being discussed is apparently Amo Bishop Roden.
"1969, in the sunshine" (additional vocal heard from 3:20 onwards). enhanced sound clip [sibobbler]