Asbury was one of the first two bishops of the Methodist Episcopal Church, which became the United Methodist Church, today the second biggest Protestant denomination in the US behind the Southern Baptists. He has a pretty interesting story, beginning with the fact that his mother had a dream that her son would be a great Christian leader. He was born in England and was in the US during the Revolutionary War, where he was the only Methodist minister in the colonies at the time.
He traveled all over the country, averaging about 6,000 miles a year and preaching where ever he could, including along the frontier of the time. He became quite popular and earned the nickname "Prophet of the Long Road," which is also inscribed in the statue. John Wesley named him the head of the Methodist Episcopal Church of the USA with Thomas Coke and under them, the church went from 1,200 members to 214,000, or about 2.5% of the total US population at the time. Lots of things were named in his honor, including schools, colleges, and Asbury Park, New Jersey, which you may know if you're from New Jersey or a Bruce Springsteen fan.
The statue on 16th Street (which has a Wikipedia page of its own) was completed by Augustus Lukeman in 1921. It was funded by the Francis Asbury Memorial Foundation and Congress approved it in 1919, as it sits in Federal land (most triangles in DC are Federal land.) Lukeman was a prominent sculptor of the time, he also worked on Stone Mountain in Georgia, the big Confederate monument.
Along with the text with Asbury's name, the statue has other inscriptions. On the left: "HIS CONTINUOUS JOURNEY THROUGH CITIES VILLAGES AND SETTLEMENTS FROM 1771 TO 1816 GREATLY PROMOTED PATRIOTISM EDUCATION MORALITY AND RELIGION IN THE AMERICAN REPUBLIC; Act of Congress," On the right: "IF YOU SEEK FOR THE RESULTS OF HIS LABOR YOU WILL FIND THEM IN OUR CHRISTIAN CIVILIZATION," and on the back: "THE PROPHET OF THE LONG ROAD".
UPDATE: Here's Pres. Calvin Coolidge at the dedication, and the speech he gave.
First photo by Mr. T in DC
Second via Library of Congress and NCinDC