At most jobs, an employee is allotted a certain number of sick days (maybe 10) in a year. This is the total number of days that he is allowed to call in, say, "I'm sick" and get an excused absence from work. This phrase, "I used up all my sick days... So I called in dead" is a joke; the employee is saying that he has used all his allotted sick days, so he is calling work to say, "I'm dead - that's why I'm not at work today."