"At presentation" means "when a symptom/diagnosis is observed/noted by a medical professional".
In the paper you cite, they are doing microarrays on biopsied tumor tissue. "At presentation" here effectively means "at biopsy", at which time that a patient is diagnosed with hepatocellular carcinoma. It's important to note that their sampling is based on presentation time because tumor age at presentation is not going to be the same for everyone - some patients may "present" with symptoms and be diagnosed when a tumor is relatively new, others may present when a tumor is advanced. Presentation is both a convenient and clinically relevant time point - it's the earliest time a prognosis can be offered.