Picture a school building that belongs to the streets around it. The people who study there live nearby, the money that keeps it open often comes from nearby, and its doors open for local adults too, not only for its own students. That is the whole of the word community here. What changes between the two meanings is only the age of the students and the level of the teaching: in the US it is a two-year college for adults after school, and in the UK it is a secondary school for teenagers that also serves the grown-ups next door.
Wiktionary notes that the name comes from the fact that these colleges mainly attract and accept students from the local community, and are often supported by that community through property taxes.
Wiktionary 的詞源說明指出，這個名稱來自這類學校主要招收當地社區的學生，而且常由當地社區透過房屋稅支持。所以 community 指的是「誰來讀、誰出錢」，而不是課程的內容。
Wiktionary, entry 'community college' (packet evidence, source_ref kaikki:3f6f69a190036ae4:1470910032:4226)
Two branches
美國：離開中學後的兩年制學院
同一個「屬於在地人的學校」畫面，放在高中之後：學費低、門檻低，念完兩年拿證書或副學士，也可以把學分帶進四年制大學。
noun · B2
In the US, a local college with two-year courses. Students go there to learn a job skill and get a certificate or an associate's degree, or to do two years of study first and then move on to a four-year university.
社區學院（美國兩年制）
英國：兼辦社區課程的中學
同一個畫面放在中學這一層：學生是住附近的青少年，而放學後校舍再開給社區的大人使用。
noun · B2
In the UK, a secondary school that teenagers from the area around it go to, which also opens its rooms in the evening and runs classes and activities for adults who live nearby.
社區中學（英國）