Green’s Dictionary of Slang

(US) persuasive.

1873		Overland Monthly (CA) Feb. 113: So Smith had a medicine-talk with him, and tried to coax him, but the soft-soap dodge wouldn’t work [HDAS].
1907		S.E. White Arizona Nights 135: He had the medicine tongue! Ten days later him and me was occupin’ an old ranch fifty mile from anywhere.