Angell was born in Providence, Rhode Island, to Henry and Mae (née Cooney) Angell. He received a bachelor's degree in English Literature from Providence College. He married Lynn Edwards on August 14, 1971. Soon after Angell entered the U.S. Army upon graduation and served at the Pentagon until 1972. He then moved to Boston and worked as a methods analyst at an engineering company and later at an insurance firm in Rhode Island. His brother, the late Most Rev. Kenneth Angell, was a Roman Catholic prelate and former Bishop of Burlington, Vermont.

Angell moved to Los Angeles in 1977. His first script was sold to the producers of the Annie Flynn series. Five years later, he sold his second script to Archie Bunker's Place. In 1983, he joined Cheers as a staff Writer. In 1985, Angell joined forces with Peter Casey and David Lee as Cheers supervising producers/writers. The trio received 37 Emmy Award nominations and won 24 Emmy Awards, including the above-mentioned for Frasier, as well as an Outstanding Comedy Series Emmy for Cheers, in 1989, which Angell, Casey, Lee and the series' other producers shared, and Outstanding Writing/Comedy Emmy for Cheers, which Angell received in 1984. After working together as producers on Cheers, Angell, Casey and Lee formed Grub Street Productions. In 1990, they created and executive-produced the comedy series Wings.

The second episode of Frasier to air after the attacks, "Don Juan in Hell: Part 2", airing on September 25, 2001, ended with the memorial tribute, "In loving memory of our friends Lynn and David Angell". "Goodnight, Seattle", the series finale which aired May 13, 2004, featured the birth of Niles and Daphne's son who is named David in tribute.

In 2004, The Angell Foundation of Los Angeles, California awarded Providence College a gift of $2 million for the Smith Center for the Arts.