The small part is most likely a star nut
This pic is proper orientation Top = Top
If so you must install this in the end of the handle bar with a hammer. First put the screw into the top of the nut and line the nut up with the open end of the haldle bar. Lightly tap the nut into the end of the bar until the screw head is ammost flush with the end of the bar. Remove the screw with allen wrenches, install grips, put screw through plastic end cap and thresd into star nut inside the bar.
Hope this helps.
The only thing I can see going wrong is the wrong star nut for your diametre of handle bar.
Assembly is as follows: Star nut first into bar,
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