How do I fill the current selection with color?

All you must do in GIMP is use the Fill Bucket tool, holding down shift will toggle between the 'fill similar color' and 'fill whole selection' options.

Just change the foreground or background color to the color of your choice and then just drag it to your image, any part will do because whenever there is a selection on an image, any action done on the image is assumed to be on the selection. Dragging the color to any part of your image, selected or not, will fill the selection with that color.

You can fill the current selection with either the foreground color or the background color from the Edit menu. Ctrl + , and Ctrl + . respectively on the keyboard.