Why does a belief network need to be represented using a directed acyclic graph (DAG)?
Yes, we use DAGs to represent dependency relationships.

We need directed graph because condition probability P(A|B) is not same as P(B|A).

We assume it to be acyclic to get certain properties and ease calculations. You can create cyclic graph and you are going to get a lot of contradictions.

If you ease both the restriction(directed and acyclic), we get more general model called Markov Network.