As native English speakers I think we tend to relate pretty easily with the expression 時間が空{あ}いている or 時間を空ける, meaning to have free time or to make/spare time. The meaning in question is similar but not exactly the same. The も and the lack of the direct object kind of obfuscates the real meaning. Indeed, it could be read as 3週間を空けた, but that would mean he essentially cleared up 3 weeks to be less busy. Instead it's referring to the website/place where he usually is: 3週間(ここを)空けていた. This usage refers to being away from somewhere. Literally it means that he "emptied" the site, referring to his not being there figuratively.
On alc we see some examples of this:
_日間家を空ける
be away from home __ days
何も聞かれずに一晩中家を空ける
stay out all night with no questions asked
10分間部屋を空ける
leave a room for __ minutes
Hopefully this is clear! It's just a literal idea of leaving a place vacant.