Falla was born in Palmerston North in 1901 to George Falla and his wife, Elizabeth Kirk. As his father was working for the railways, the family shifted frequently, and he lived in Hawera, Masterton, and Invercargill. At his primary school in Invercargill, he developed an interest in natural history due to the influence of Alfred Philpott. Falla gained a junior national scholarship and studied at Auckland Grammar School, from where he graduated in 1918. He pursued his dream of going to sea for a few years in various jobs, but then started to follow his interest in ornithology, first through part-time study at Auckland University College, and then at Auckland Training College, a training institution for teachers. He transferred to the university in Auckland, from where he graduated in 1924 with a Bachelor of Arts, in after some teaching at primary schools, with a Master of Arts in 1927.
He was assistant zoologist with the British, Australian and New Zealand Antarctic Research Expedition (BANZARE) under Sir Douglas Mawson 1929–1931. He was the founding president of the Ornithological Society of New Zealand. He was involved in the organisation of the wartime subantarctic Cape Expedition coastwatching program of 1941–1945.
He held positions in various New Zealand museums including director of Canterbury Museum from 1 March 1937 to 1947, and director of the Dominion Museum, Wellington, 1947–1966. He was a member of the Royal Australasian Ornithologists Union (RAOU), president 1951–1952, and made a fellow of the Royal Australasian Ornithologists Union in 1973.
On 18 May 1928, Falla married Elayne Mary Burton, known as Molly, at Te Aroha; they were to have two daughters and one son. His wife died in May 1978. Falla was found dead at his home in the Lower Hutt suburb of Eastbourne on 24 February 1979.