hyudaeponeul jibeseo chungjeonhae dugo nawassneundedo baeteoriga ppalli jureo.

휴대폰을 is the object of 충전하다 (to charge): you charge the phone. The second clause switches to a new subject: 배터리가 (the battery) is what 줄어 (decreases/drains).
So the sentence has two different “roles”:

• 휴대폰을 → object of 충전해 두다 (charged it)
• 배터리가 → subject of 줄다 (drains)

충전해 is the informal connective form of 충전하다:

• 충전하다 → 충전해 (하다-verbs go to -해)

Here it’s followed by 두고 (from 두다), forming 충전해 두고. That chunk is basically “charge (it) and leave it that way / have it charged in advance.”
It’s not just “charge and then…”—the 두다 adds the nuance of preparing something and keeping it in that completed state.

-아/어 두다 means do something in advance and leave it done (often for later convenience), or do it and keep the result/state.

• 충전해 두다 = to have it charged / to charge it and leave it charged It often implies “I made sure it was done before I left / before it was needed.”

두고 is 두다 + -고 (and/then), linking actions:

• 충전해 두고 = “having charged it (and leaving it charged), and then…” It sets up the next action 나오다 (to come out/leave).

-는데도 means even though / despite the fact that.

• 나왔는데도 = “even though (I) left / came out”

Structure-wise:

• [Background fact + -는데도] + [unexpected result] So: “Even though I left after charging it (so it should be fine), the battery still drains quickly.”

Both can translate as but / although, but the feel is a bit different:

• -는데도 strongly highlights unexpectedness or “in spite of that fact”
• -지만 is a more neutral contrast

So 나왔는데도 배터리가 빨리 줄어 implies “I did what should help (charged it), but surprisingly it still drains.”

Korean often omits the subject when it’s obvious from context. Here, it’s naturally understood as I:

• (내가) 휴대폰을 집에서 충전해 두고 나왔는데도… In everyday speech, adding 내가/나는 would often sound unnecessary unless you’re emphasizing “I (not someone else).”

줄어 is the informal present form of 줄다 (to decrease). Here it means the battery level decreases/drains.

You’ll also commonly hear:

• 배터리가 빨리 줄어들어 (same meaning; 줄어들다 = “to decrease/go down”)
• 배터리가 빨리 닳아 (닳다 = “to wear down,” commonly used for batteries)

집에서 marks the location where an action happens:

• 집에서 충전하다 = “charge it at home”

집에 often marks destination or location-existence:

• 집에 가다 (go home)
• 집에 있다 (be at home)

Since charging is an action performed in a place, -에서 fits best.

That mix is normal:

• 나왔는데도 is past because leaving already happened (“even though I left…”)
• 줄어 is present because it’s describing what’s happening now / a current ongoing tendency (“the battery drains quickly”)

So it reads like: “I charged it before and left, but (now) it still drains fast.”