Born in Farmington, Utah Territory to George F. Richards and Alice Almira Robinson, his father served many years in the Quorum of the Twelve. As a young boy, LeGrand had several accidents that could have taken his life: once as a small child he was struck in the head by an ax as he approached his father from behind while his father was chopping wood. A few years later LeGrand was thrown from a wagon by an agitated horse and both the wagon wheels rolled over his head. As a child Richards attended the 1893 dedication of the Salt Lake Temple. Richards's church service began when he filled a proselyting mission to the Netherlands between 1905 and 1908.

Richards returned to the Netherlands as the presiding elder over the mission, accompanied by his wife Ina Jane Ashton Richards, from 1914 to 1916. Richards was ordained a high priest and bishop on June 29, 1919, by Charles W. Penrose, and presided over a Salt Lake City ward from 1920 to 1925. In 1926, he filled a short term mission primarily serving in Rhode Island. In about 1930 Church President Heber J. Grant sent Richards to southern California with the plan to call him as stake president. However the then-current stake president called Richards as a bishop and convinced Grant to hold off on calling him as the stake president so local members would not feel as if Richards was an outsider being imposed on them. Between 1931 and 1933, Richards presided over the Hollywood Stake.

In 1933 and 1934 Richards again lived in Salt Lake City, where he served on the stake high council of the Liberty Stake. He was called to this position by stake president Bryant S. Hinckley. Following this Richards served as president of the Southern States Mission from 1934 to 1937. He was called to this position to replace Charles A. Callis who had been called to the Quorum of the Twelve Apostles.

Richards served as presiding bishop during and after the Second World War and began to adopt building programs to deal with the massive post-war growth in membership of the church.

Outside of his apostleship, Richards is probably best known for his widely distributed book, A Marvelous Work and a Wonder: Church of Jesus Christ of Latter-day Saints, which was first published in 1950. The commonly referenced work contains a comprehensive teaching outline designed to assist missionaries in their study and presentation of Mormonism. Based on a document titled The Message of Mormonism Richards developed in 1937 for missionaries during his tenure as president of the Southern States Mission, it contains explanations and interpretations of many doctrinal positions of the LDS Church.

In 1955 Richards published Israel! Do You Know?, an effort to demonstrate the links between Jewish traditions and beliefs and Mormonism; this document was produced in conjunction with an LDS Church program aimed at proselyting Jews living in Southern California.

Richards also played a role in LDS connections with Israel. He was the head of the Orson Hyde Foundation, which coordinated most of the money sent to Jerusalem which was donated to that city in exchange for the land that became the Orson Hyde Memorial Garden.

In a memorial address read by his personal secretary after Richards's death, Church President Spencer W. Kimball paid tribute to Richards as

Richards was the longest-lived LDS Apostle until David B. Haight; Both his father George F. Richards and grandfather Franklin D. Richards had served as President of the Quorum of the Twelve Apostles. Franklin D. Richards was also the nephew of Willard Richards, another Apostle and notable leader in LDS church history.