How do the departments and agencies of the executive branch work to implement the law?

Generally, what the agencies in the executive branch do is
take the laws and do two things.  First, they make the laws more specific by writing
rules that are based on the laws.  Second, they enforce those rules in a variety of
ways.

When Congress writes laws, it does not tend to make
them very specific.  It will set general goals (get students up to grade level by year X
for example) and leave it up to the bureaucrats to write rules specifying how this must
be done.

The other thing that bureaucrats do is they
enforce the laws.  They may, for example, go out checking on businesses to make sure
that they are following the laws in terms of worker
safety.

So the executive agencies tend to write rules to
flesh out laws and they tend to enforce the laws and rules.  By doing these things, they
implement the laws.