What does 开 mean in 会开完了 ("the meeting is over")?
This is a good example of a V.O. (verb + object) word and it’s flexibilities.
If we take a look at Tuttle Learners Dictionary’s entry for 开会, we have;

[v+obj: 开 open up + 会 meeting]
VERB
attend a meeting, hold a meeting

and example sentences:

我们最好开个会, 讨论一下这个问题。
Wǒmen zuìhǎo kāi ge huì, tǎolùn yíxià zhège wèntí.
We'd best have a meeting to discuss this issue.
王老师在开会。
Wáng lǎoshī zài kāihuì.
Teacher Wang is at a meeting.

The order of V.O. words can be switched around. In your example for instance it starts with the object and then uses the verb later.
You might see similar expressions like:

话说完了

饭吃完了

字写完了