Learn Hardcore Korean: jega johahaneun pyohyeoneun “gamsahapnida”yeyo.

In the phrase 제가 좋아하는 표현, 제가 uses the subject marker -가, which is required inside a relative (noun-modifying) clause. Here you’re literally saying “the expression that I like.” If you tried 저는 좋아하는 표현, you’d be marking “I” as the topic of the main clause, which doesn’t work when you want to directly modify 표현.

What does 좋아하는 표현 mean, and why is 좋아하는 placed before 표현?

Korean uses verb stems in an attributive form to modify nouns.

좋아하다 = “to like”
좋아하는 = “that (someone) likes” (attributive form)
So 좋아하는 표현 literally means “the expression (표현) that (I) like.” Placing 좋아하는 before 표현 turns the verb into an adjective for that noun.

Why is 표현 followed by the topic marker -은 (as in 표현은)? Could you use -이/-가 instead?

-은 marks 표현 as the topic of the main clause: “As for the expression I like…”
If you used 표현이, you’d be marking it as the subject, which doesn’t flow naturally into -예요 here. The topic marker lets you introduce 표현 and then comment on it (“it is ‘감사합니다’”).

Why does the sentence end with 예요 rather than the more formal 입니다?

Both are polite copulas meaning “is/am/are.”

예요: polite, everyday speech (lower formal)
입니다: polite, formal writing or speeches (higher formal)
Using 예요 makes the tone friendly and conversational.

Why is 감사합니다 shown in bold quotes here? Is that necessary?

Because the speaker is naming a specific expression. Quotation marks (or bolding) highlight 감사합니다 as the exact phrase they like. In Korean you often see:
제가 좋아하는 표현은 “감사합니다”예요.
Without quotes it’s still correct, but the quotes clarify that you’re naming the phrase itself.

What’s the difference between 감사합니다, 감사해요, and 고맙습니다?

All three mean “thank you,” but differ slightly:

감사합니다: formal polite (Sino-Korean root 감사 + formal ending ㅂ니다)
감사해요: polite casual (same root + 해요 ending)
고맙습니다: formal polite (native Korean root 고맙다 + ㅂ니다)
Use 감사합니다 or 고맙습니다 in formal situations; 감사해요 is softer and more casual.

Could you replace 표현 with 말 here? Would that change the nuance?

Yes. 말 means “word/what one says,” while 표현 means “expression/phrase.” In this context they overlap:
제가 좋아하는 말은 감사합니다예요.
sounds perfectly natural. 표현 just feels a bit more general or abstract.

Can you omit 제가 if it’s clear who you’re talking about?

In casual or spoken Korean, you can drop subjects when context is obvious:
좋아하는 표현은 감사합니다예요.
However, including 제가 is clearer—especially in writing or when introducing a new topic.