The phrase isn't literal; it means the speaker's voice is not clear, not that an actual frog is present. Sorry, I can't speak clearly right now - I have a frog in my throat.
Pro Tip 2/3
Temporary Discomfort
Use this phrase to indicate a temporary hoarseness or difficulty to speak that you expect to pass. Can we reschedule the call? I've got a frog in my throat this morning.
Pro Tip 3/3
Informal Use
The phrase is informal, more suitable for casual conversation than professional or formal situations. During the family dinner, Uncle Joe apologized for his raspy voice, saying he had a frog in his throat.