Learn Hardcore Korean: tellebijeoneseo misulgwan jeonsihoereul sogaehaneun nyuseureul bwasseoyo. - I watched a television news segment introducing the art museum exhibition.

Because 미술관 전시회 is the direct object of the verb 소개하다 (“to introduce”). Korean marks a direct object with -을/를. Here, vowel-final 전시회 takes -를, giving 미술관 전시회를 (“the art museum exhibition”).

How does 소개하는 function in 미술관 전시회를 소개하는 뉴스?

소개하는 is the present-tense attributive (or adnominal) form of 소개하다. In Korean, a verb that modifies a noun (a relative clause) must use its attributive ending, not the sentence-ending ending.

소개하다 + 는 → 소개하는 (“introducing”)
This whole clause (미술관 전시회를 소개하는) comes before the noun 뉴스, giving “news that is introducing an art museum exhibition.”

Why do we use 보다 with 뉴스를 to mean “watch the news”?

In Korean, 보다 covers both “to see” and “to watch.” Common collocations include:

영화를 보다 (“watch a movie”)
드라마를 보다 (“watch a drama”)
뉴스를 보다 (“watch the news”)
So even though English distinguishes “see” vs. “watch,” Korean simply uses 보다.

Why is there no explicit subject in this sentence?

Korean often omits the subject when it’s understood from context. Here, the speaker (“I”) is obvious, so 저는 (“I am”) is dropped. The full version would be:
저는 텔레비전에서 미술관 전시회를 소개하는 뉴스를 봤어요.
But omitting 저는 is more natural.

What does 봤어요 tell us about tense and politeness?

봤어요 is the past-tense polite form of 보다.

The -았/었어요 ending marks past tense in the 해요 style (polite, informal).
So 봤어요 means “(I) watched” or “saw,” politely.