Welcome to the (surprisingly large) group of '59 Tradewind owners. You will find a lot of info by using the search function to review topics in common to all of us.
The 12 volt
battery originally was housed in a rectangular box beneath the front window. Wiring entered the trailer on curbside, behind the front gaucho, and from there was distributed to the trailer. This battery box was/is much smaller than a standard 12 volt
RV battery today. A previous owner of your trailer must have used a newer, standard sized RV battery that was housed under the sink compartment, and there attached the "battery cables" you have into the trailer's 12 volt
system. If you do this, be careful to vent the battery's gasses to the exterior of the trailer.
Also behind the gaucho in front is the fresh water tank - an aluminum cylinder approximately 16" in diameter and 5 feet long. The interior of mine was heavily coated with white sediment, and I discarded it. Others have been able to use the original tank, however.
Brad, aka "flyfshr" on this list, maintains a registry of '59
Tradewind owners. I'd expect that he will contact you to obtain more info about your trailer. His trailer is pictured on the Vintage Airstream photo archives at the example of how good a '59 Tradewind CAN look:
Please send along some more photos, and also the trailer's VIN number for the registry.
Best of luck with your new trailer!