여쭈다, appearing here always as 여쭤보다, is the humble word for asking: you 여쭈다 someone above you and 물어보다 anyone else. The -게 adverb says how the asking is done, and in these lines it is about manner and permission — 조심스럽게 'cautiously', 정중하게 'respectfully', 편하게 여쭤보시면 'do feel free to ask'. It turns up constantly in interviews, where the host is marking deference to the guest.