While the "Floating Selection" is a bit odd at first, it is actually a nice feature, because it allows you to manipulate the floating selection as much as you like before anchoring it to the layer you pasted it too. So, a step by step walk through the process of copying and pasting selections from one layer to another layer in GIMP would be as follows:

1. Make the layer you want to copy from the active layer (click on that layer);
2. Use your Free Select tool to create a selection of the desired area;
3. Right click on the image and go to: Edit > Copy;
4. Make the layer you want to paste to the active layer;
5. Right click on the image and go to: Edit > Paste (this will create a "Floating Selection: Pasted Layer" above that layer);
6. Manipulate the pasted layer if you so desire (e.g. rotate it, resize it etc);
7. In the Layers Dialogue, click on "Anchor Floating Selection".

That should do it.

Cheers,
-Arsheesh