Fallout 76 is the fifth series of the Fallout game, after the previous series, Fallout 4, released in 2015. There is no clear explanation of the reason Bethesda picked the number 76, but all the pertinent puzzles can be found on the new teaser trailer video. The number 76 itself has many interpretations.
At the beginning of the video, a song played on the old radio called "Take Me Home, Country Roads" by John Denver in 1971. On the lyrics of the song is mentioned a location in the United States called West Virginia. More details there is a date listed on the device, ie, October 27, 2102.
Slowly the video shows a neatly arranged Vault and was shown a video recording on television featuring a moment of Vault 76 inauguration. As is known Vault is a shelter of protection in this game, populated by the so-called Dweller.
Next, the video again highlights the condition of Vault that appears to be Vault 76, it appears that the Vault is like just celebrating something. On one of the banners attached written Reclamation Day which seems to be the event of the celebration.
Toward the end of the video one Dweller with a typical blue look puts the Pip-Boy device and the video highlights the Dweller's back number of 76.
Most likely, Fallout 76 will be a major attraction of the Bethesda conference at E3 2018 on 11 June 2018.