Complex sentence: Both versions have eating make the horses calmer, and Heracles took the opportunity to bind their mouths shut, and easily took them back to King Eurystheus, who dedicated the horses to Hera.
Split into simpler sentences: Both versions have eating make the horses calmer, and Heracles taking this opportunity to bind their mouths shut. Once subdued Heracles was easily able to take them back to King Eurystheus who dedicated the horses to Hera.