Learn Finnish: Me väitimme, että hän on sairas.

Väittää (to claim) is a Type 1 verb. To form the past tense for 'we', we add the past tense marker -i- and the personal ending -mme. Because -mme closes the syllable (väi-tim-me), it triggers consonant gradation. The strong tt weakens to a single t.

Why does the sentence use on (is) instead of the past tense oli (was)?

In English, we often match the tenses in reported speech (We claimed that he was sick). Finnish does not do this! You keep the exact tense of the original statement. Since the original claim was 'He is sick' (Hän on sairas), Finnish keeps the present tense on in the second part of the sentence.

Is the comma before että mandatory?

Yes. In Finnish, you must always place a comma before the conjunction että (that) when it separates two clauses (parts of the sentence that each have their own subject and verb).

Could I drop the pronouns and just say Väitimme, että on sairas?

You can drop me because the -mme ending on väitimme already tells us the subject is 'we'. However, you cannot drop hän. In Finnish, third-person pronouns like hän (he/she) and he (they) must always be included in the sentence.