Grassby was born in London and emigrated to Canada in 1902. Primarily a wood engraver, He spent several years in various locations in the United States, went to Montreal in 1913, and settled permanently in Boston shortly thereafter. Grassby worked both as a successful typographic designer and wood engraver, producing pictorial engravings for various commercial clients, calendar illustrations, and decorative engravings of Boston area landmarks. A major collection of his work is housed in the Boston Public Library.