Learn Hardcore Chinese: jīntiān wǒ hěn máng,tài máng le.

In Chinese, many adjectives (like 忙) are stative verbs and can serve as the predicate by themselves. For a neutral description you usually add a degree adverb like 很 (hěn) before the adjective: 我很忙 = “I’m busy.” Using 是 before a bare adjective is generally wrong in neutral statements. You only see 是 with an adjective in a contrastive/emphatic frame, e.g., 我是很忙，但是还是会去 (“I am busy, but I’ll still go”).

Does 很 (hěn) here actually mean “very”?

Not necessarily. In simple statements like 我很忙, 很 often functions as a neutral linker to make the sentence sound natural, and doesn’t strongly mean “very.” If you really want to emphasize “very,” use 非常, 特别, 真的, or stress 很 in speech.

Can I drop 很 and just say 我忙?

Grammatically possible but less natural as a neutral statement. 我忙 can sound abrupt, contrastive (“I’m busy [as opposed to free]”), or elliptical (like answering a question: “忙！”). For everyday statements, 我很忙 is safer.

What does 太…了 mean in 太忙了?

太 (tài) … 了 (le) expresses an excessive or extreme degree, often with emotion/complaint: “too/extremely …!” Here 太忙了 = “(I’m) too busy!”