The slave trade in Washington DC had been a contentious issue for abolitionists and anti-slavery advocates for years. The District of Columbia was unique in that it did not have its own local government in the way that the states did. Their government was the Congress of the United States, and therefore were part of a unique jurisdiction. While Congress could make no laws respecting slavery in the states, they did have the authority to abolish the slave trade in DC.

In the 1830s and 1840s, the House of Representatives was flooded with petitions from across the "free states" asking for the abolition of slavery or the slave trade (two distinct sets of petitions) in the District of Columbia. Due to the Gag rule in effect in the House, the petitions were immediately tabled and could not be discussed. This became a hot issue for both the Representatives and their constituents, because the "slave power" had now managed to restrict the right of citizens to petition their government. The right to petition the government was enshrined in the first amendment, and the issue of the "gag rule" united anti-slavery advocates with more moderate supporters across the north (Fredrick, “John Quincy Adams, Slavery, and the Disappearance of the Right of Petition”, Law and History Review (Spring 1991) pp. 113-155; Miller, Arguing about Slavery)

Lastly, the petitioners against the slave trade in the District of Columbia had two major arguments they used in their petitions. Firstly, that they considered the slave trade to be an embarrassment for the country, in that Congress would meet and discuss the freedom of the people, while across the street a slave auction would take place. Ambassadors from England in particular found this juxtaposition shameful and supported the abolitionists. In one example the petitioners quoted the Marquis de Lafayette, during one of his visits to America, calling slavery “a dark spot upon the character of the nation" (Petition from the inhabitants of Wayne County, Michigan presented January 23, 1837). The second argument was that the institution of slavery was unchristian (a common anti-slavery position). But the petitioners went further, and either warned congress or expressed fear that God would punish those who allowed slavery to exist. They rarely extended this to an argument that God's law was higher than man's law, and instead stuck to an attack on the slave trade in DC itself, as that was within man's law to execute (Walters, Anti-Slavery Appeal; Petitions from the male citizens of Pennsylvania, Presented Jan 23, 1837).

Thus, in order to help appease the Northern states into accepting the Compromise of 1850, and especially it's new harsher fugitive slave law, Clay and others included the abolition of the Slave Trade in DC (Potter, Impending Crisis).

EDIT: Here is an example of one of the thousands of petitions sent in in the 1830s.