Translation:He caught a cold last week.
Well... Before getting a cold you are usually fine, so there IS a change of state. Do not think in English, think the Chinese way! It relies way more on context than English does.
下雨了 It's raining (It was not raining a minute ago/one hour ago / yesterday) 昨天下雨了 It rained / was raining yesterday
生病了 I'm sick 昨天生病了 I was sick yesterday / I got sick yesterday / I had a cold yesterday. I still might be sick today, or maybe not.
Chinese relies on context to convey more precise information. Don't think in English. 了 here indicates a change of state, it does not give any clue about it's completion. Well, technically speaking, the change itself happened and has been completed, but we don't know about the new state this change led to. I'm not sure I'm clear haha...