-(으)러
-(eu)reo
The feet move first: -(으)러
The picture Putting one errand in your bag and stepping out the door. The purpose is the load; the body carries it somewhere. -(으)러 only attaches to a trip you actually make, while -(으)려고 needs nothing but a target in your mind — 약 사러 가요 works, 약 사러 돈을 모아요 does not; it has to be 약 사려고 돈을 모아요.
Where the picture shows up in -(으)러
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the errand you carry out
점심 먹으러 갑시다.
jeom-sim meo-geu-reo gap-si-da
Let's go get lunch.
Getting up from your desk, picking up nothing but your wallet, and heading for the office door.
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why you went that far
친구 만나러 홍대에 갔어요.
chin-gu man-na-reo hong-dae-e ga-sseo-yo
I went to Hongdae to meet a friend.
Coming up out of the Hongdae subway exit and raising a hand to the friend who was waiting.
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a quick trip out and back
약 사러 약국에 다녀올게요.
yak sa-reo yak-gu-ge da-nyeo-ol-ge-yo
I'll run to the pharmacy for some medicine.
Slipping a single prescription slip into your pocket and stepping out for a moment.
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what brought you here
택배 찾으러 왔는데요.
taek-bae cha-jeu-reo wan-neun-de-yo
I'm here to pick up a parcel.
Standing at the front-desk window, holding up the screen with the tracking number on it.
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a trip you keep making
매주 수영하러 다녀요.
mae-ju su-yeong-ha-reo da-nyeo-yo
I go swimming every week.
Shouldering a swim bag and walking in through the same pool door again, week after week.
Only verbs of movement — 가다, 오다, 다니다 — may follow -(으)러. 먹으러 공부해요 is wrong.
All of it in one picture
-(으)러 is putting one errand in your bag. Take 'lunch' as your load and your feet head for the office door; take 'meet a friend' and your feet go all the way to Hongdae; take 'buy medicine' and your feet go to the pharmacy and back. Take 'pick up a parcel' and it becomes what brought you here; pick the 'swimming' load up again every week and the trip turns into a habit. That is why -(으)러 is followed by verbs of real movement — 가다, 오다, 다니다; with a load but no movement, -(으)러 has nothing to ride on.