Learn French: Ils ont ajouté qu'ils étaient au chômage. - They added that they were unemployed.

This is the rule for reported speech. The main verb, ont ajouté (added), is in the past. When the people originally spoke, they would have used the present tense: "Nous sommes au chômage" (We are unemployed). Because we are reporting this later using a past tense verb, the present tense sommes "backshifts" (takes a step back in time) into the imparfait étaient.

Why do we use the phrase au chômage instead of an adjective for "unemployed"?

In French, being unemployed is usually expressed with the noun phrase être au chômage, which literally translates to "to be at the unemployment". While there are related adjectives in French, using this phrase with être is the most standard and natural way to say someone is out of work.

Why did the pronoun change to ils in the second half of the sentence?

When shifting from direct to reported speech, the pronouns have to match the new perspective. Originally, the speakers would have said "Nous sommes..." (We are...). Since we are now talking about what they said, the "we" becomes ils (they).

Why is it written as qu'ils instead of que ils?