Learn Hardcore Korean: uri jibeun yeogirang gakkawo. - My place is close to the station.

은 is the topic marker. It sets “our house” as the topic of conversation: “As for our house, …” It can also carry a slight contrastive nuance (“our house (as opposed to others) is close to the station”).

Using 이 instead (우리 집이) would mark “our house” as the grammatical subject and often feels like you’re identifying or selecting it (“It’s our house that’s close to the station,” e.g., in answer to “Which house is close?”).
Both are possible; 은/는 is about what you’re talking about, 이/가 is about what’s doing/being in the sentence or what’s being singled out.

Why is there 이랑 after 역? Is that the subject particle 이?

Here, 이랑 is a single particle meaning “with/and,” used colloquially to express a relation, including spatial ones. It’s not the subject marker 이. Because 역 ends in a consonant, you use 이랑; after a vowel you’d use 랑 (e.g., 학교랑).

Pattern: [Topic/subject] + [reference point]+(이)랑 + 가깝다 → “X is close to Y.”
More neutral alternatives: 하고 (speech), 와/과 (formal/written).

Can I say 역에 가까워 or 역에서 가까워 instead? What’s the difference?

Yes:

역에 가까워요 = “[It] is close to the station.” (Direct “to” relation; concise and standard)
우리 집에서 역이 가까워요 = “From our house, the station is close.” (Vantage point is ‘from the house’)
우리 집은 역에 가까워요 is very natural too. 이랑 is more casual; 에/에서 are more textbook-neutral.

How polite is 가까워? What should I say to a stranger, teacher, or in writing?