where does the little spring go inside the carburetor on Ryobi 15" Electric Trimmer?

The little spring inside of the carburetor goes under the float arm.
That is where your fuel inlet needle/float valve is located...on the arm at the end.
Normally there is a small indetion in the carburetor base and a small protrusion on the underneath of the float arm where the spring will be in the right postion for installation.
The spring will set in the indention and you will install the float arm with the needle/float valve and float rod into position over top of the spring,you will push down until it is in position and then you can tighten the screw that holds the float arm assembly in position.
Your carburetor should be working properly to run now as long as you do not have other problems in the fuel system or electrical problems.