What would possess a man to want more than one wife?

I would think one would be more than enough.
In our society, there's not really any good reason for it. IMHO.
But in the past, and in other societies - especially agrarian, there were/are cultural and economical reasons for it.
Multiples wives means more children and a better chance of some children living to adulthood.
Childbirth used to be the leading cause of death for women (still is in some places); so multiple wives provided stability. If one died, the others could raise her children, do her chores, etc. The husband wouldn't be in a position where he'd have to look for and quickly find someone to do those "womanly" things - the structure would already be in place.
Multiple wives worked as a sort of co-op for child care and other "female" chores. While one wife would be hard pressed to take care of young children
and get the laundry done or the cooking or whatever, two wives could split the work - instead of them both being burdened by young children, one could look after them while the other did other work. And three wives meant even more work done. A household could in that way be self-sufficient.
In agrarian and nomadic societies this structure was beneficial because neighbors were few and far between and the work was harder and took longer. Having a wife or a concubine was more economically feasible than hiring a servant or buying a slave (although, oftentimes there wasn't much difference between buying a female slave and buying a wife).