-(으)ㄹ게요
eul-ge-yo
A stamp pressed in front of the listener: -(으)ㄹ게요
The picture Stamping one thing you will do and handing it to the person in front of you. -(으)ㄹ게요 is a promise handed to someone; -(으)ㄹ래요 is what your own mind has chosen.
Where the picture shows up in -(으)ㄹ게요
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a stamp on calling back
제가 다시 전화할게요.
je-ga da-si jeon-hwa-hal-ge-yo
I'll call you again.
Before hanging up, pressing 'I will call back' onto a slip and putting it in their hand
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a stamp on a deadline
내일까지 보내 드릴게요.
nae-il-kka-ji bo-nae deu-ril-ge-yo
I'll send it to you by tomorrow.
Tapping tomorrow's square on the calendar and stamping it where they can see
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a stamp on being on your way
금방 갈게요. 오 분만 기다려 주세요.
geum-bang gal-ge-yo. o bun-man gi-da-ryeo ju-se-yo
I'm coming. Just give me five minutes.
Already turned toward the person waiting while you speak into the phone
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a stamp on taking the job
이건 제가 할게요.
i-geon je-ga hal-ge-yo
I'll take this one.
Pulling one document on the shared desk over to your own side
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a stamp on not repeating it
다음에는 늦지 않을게요.
da-eu-me-neun neut-ji a-neul-ge-yo
I won't be late next time.
An apologetic face, stamping next time's square in advance
It needs a listener. It cannot be used for the weather or for other people: 비가 올게요 is simply wrong.
All of it in one picture
-(으)ㄹ게요 is one stamp pressed while someone watches. Press it on a phone call and you get 다시 할게요; press it on the calendar and you get 내일까지 드릴게요; press it on your own feet and you get 금방 갈게요. Press it on a document and it means the job is yours; press it on next time and it becomes an apology. A stamp always needs someone to receive it, so this form only lives when there is a listener.