Learn French: C'est un mystère, mais quelqu'un observe un indice. - It is a mystery, but someone is observing a clue.

Quelqu'un means "someone", and just like in English, it counts as a singular subject. Therefore, you always conjugate the verb in the third-person singular (il/elle/on) form. That is why we use observe here.

Why is there an apostrophe in quelqu'un?

It is a contraction of quelque (some) and un (one). Because quelque ends in a vowel and un begins with a vowel, the e is dropped and replaced with an apostrophe to make pronunciation smoother.

Why does the sentence use c'est instead of il est for "it is"?

You use c'est when "it is" is followed by a noun phrase, such as a noun with an article. Because mystère is a noun and has the article un in front of it, you must use c'est un mystère. You generally use il est when describing "it" with just an adjective.

The English translation says "is observing". Why is there no French word for "is" before observe?

French does not have a specific "is doing" structure (like the English present continuous). The simple present tense verb observe translates to both "observes" and "is observing" depending on the context.