The expression must have been in popular use in the USA by 1894, when it was used in an advertisement as the name of a proprietary medicine for sore throats, in The Stevens Point Journal, November 1894: "The Taylor Bros. say that 'Frog in the Throat' will cure hoarseness. 10 cents and box." One theory about a frog in the throat is that the secretions of frogs helped cure sore throats. But that is just a theory, most likely not true but interesting to think about!   ( source ) To say it today is to indicate that someone's voice has gone scratchy or rough.