"아들은 아빠에게 편지를 보내줄 거야."
Translation:A son will send a letter to the dad.
주세요 is the present tense of 주시다, the honorific form of 주다 (= to give), where the 'honoring' (respect) is what is felt by the Speaker towards the Subject of the sentence.
[The polite -요 ending on the other hand is directed at the Listener(s).]
주세요 can be used as an active verb, in such case it stands for give ; or as an auxiliary verb, in such case it stands for do as a favour (for) 
 as an active verb: 주세요, give
• 주세요, indicative mood
할아버지께서는 매일 우리에게 용돈을 주세요. Grandfather gives us pocket money every day.
• 주세요, interrogative mood
새로운 선생님이 주말에 할 숙제를 주세요? Does the new teacher give you homework to do over the weekend?
• 주세요, imperative mood
물 좀 주세요 Please give me some water
 as auxiliary verb: V[어/아] 주세요 Do a favor for someone by V-ing
• [어/아] 주세요, indicative mood
영어 교사가 오늘 수업 시간에 학생들에게 영화를 보여 주세요. The English teacher shows ("does a favor for the students by showing") a film to students in class today
• [어/아] 주세요, interrogative mood
어젯밤에 할아버지가 너희에게 동화책 읽어 주셨어? Did your grandfather ("do a favor for you kids by reading") read some fairy tale books to you kids last night?
• [어/아] 주세요, imperative mood
식탁을 닦아 주세요 Please ("do a favor for me by wiping") wipe the dining table
Personally, this seems to read:
아들은 The son (Son of someone both known to Speaker and his dad)
아빠 Speaker's own dad. This implies the Listener is the Speaker's father
보내줄 거야 will send on my (Speaker's) behalf.
So: 아들은 아빠에게 편지를 보내줄 거야 = The son will send for me the letter to you, dad.
～에게 ～주다 ("to" someone, 주 as an auxiliary meaning on someone's behalf) together mean for someone. "To" is not a great translation of this particle in this sense. It's the same in Japanese, even French has a similar thing going with à ("a-grave"!?). Only English seems to resist dancing "to" someone else's tune.
- 어/아 주다 in this case as you rightly point out is only an auxiliary verb which plays no active role in the meaning of the sentence apart from adding to it a degree of politeness.
The main verb here is 보내다, to send. In both languages, this verb is a double-object verb (1× direct object and 1x indirect).
에게 보내다 = to send to
In English however, the preposition 'to' is often dropped, but only when the indirect object goes immediately after the verb.
I send 'you' a reply = I send a reply 'to you'. (Ind. obj. = you) [ And not: I send a reply you (X) ]
Incidentally, "to give" is also a double-object verb. "To feed" is another example.