First... this sentence is horribly written (which might be why you are having trouble parsing it).
Argue in this case is the normal sense of the verb. The "author" is making an argument.
Let's break the sentence down...
The author is arguing in support of a "primary suggestion".
The "primary suggestion" is- "fining those who do not recycle in order to pay recycling collectors more for their work."
(I am pretty sure that the use of the gerund form "fining" is grammatically incorrect.)