饶 here means 'abundant / plenty of' (as in 丰饶). The phrase is typically used adverbially (饶有兴趣地 + verb) to describe watching or listening with absorbed, active interest — stronger than plain 感兴趣.
memory
Think 饶 = 'plenty,' 有 = 'have,' 兴趣 = 'interest' → 'has plenty of interest.' Stack the four characters literally and the meaning is almost transparent.
