This is, broadly speaking, incorrect. Slavery was not the primary motivation behind the American Revolution.

History does not occur broadly, however. Looking at specific examples at specific places, we see that some Americans did support the Revolution because they believed the British were a threat to slavery.

Looming in the background of American slaveholders' minds during the 1770s was the Mansfield Ruling. In this case, an enslaved black man made his way to England and escaped from his master, found supporters in England's nascent abolitionist scene, and sued for his freedom on the grounds that slavery did not exist in England itself. The Chief Justice ruled in the enslaved man's favor, and set the precedent that the royal government could (seemingly arbitrarily) declare whole territories slave-free zones. American slaveholders, correctly as it turns out, realized that this case could set the groundwork the eventual abolition of slavery across the British Empire. When they spoke of British threats to their property, slave property was very much on their minds.

More acutely, the governor of Virginia in 1775 was Lord Dunmore. Dunmore had no patience with the quarlsome Virginia planters he had been appointed to rule, but was impressed by the loyalty Virginian slaves seemed to show the Crown. Dunmore decided to try a risky, ultimately disastrous, game of brinksmanship: he threatened white Virginians with freeing or impounding their slaves and turning them into a militia that he could use to force compliance with Crown taxation efforts. This backfired as completely as a thing can backfire: on April 21, 1775 (two days after Lexington and Concord, though no one involved knew that) British troops seized a Patriot powder magazine, causing the Virginia militia to rise up and chase the governor out of Williamsburg to the safety of a British warship anchored at Yorktown.

Dunmore attempted to make good on his promise, forming Dunmore's Ethopian Regiment to encourage slaves to run away and fight for the King in exchange for freedom. While the Regiment was mostly wiped on by a smallpox outbreak, it confirmed white planter's suspicions: Dunmore was serious about using black troops to enforce loyalty among whites, and enslaved people would run off by the hundreds to organize into an force that could do violence to their former masters. Virginians reviled Dunmore as the worst kind of race traitor, and held up this example as proof of an unsolvable rift between colony and Crown.

I drew this answer from Alan Taylor, The Internal Enemy, which is where I suspect your professor drew his statement from.