The sales sense springs from real estate, where closing is the final step of a transaction.
In sales, it is used more generally to mean achievement of the desired outcome, which may be an exchange of money or acquiring a signature.
"Closing" is distinguished from ordinary practices such as explaining a product's benefits or justifying an expense.
For example, a salesman might mention that his product is popular with a person's neighbors, knowing that people tend to follow perceived trends.
In automobile dealerships, a "closer" is often a senior salesman experienced in closing difficult deals.