If a person really did have a frog in his or her throat, that person would have a very difficult time speaking. As an idiom, to have a frog in your throat means to be speaking with a hoarse voice.
Hoarse means gruff, deep, and harsh. It is usually caused by some kind of illness.
Example: "Mary, are you all right? You sound like you have a frog in your throat." "Ah, I'm fine, it's just a cold, nothing serious." Here, Mary's voice sounds hoarse, but it is not (to anyone's knowledge) caused by a serious illness, but by the common cold.