1 – The original image
Select the Freeform Pen tool from the Toolbox (click and hold down on the pen tool if it's not immediately visible.)
2 - Creating a Path
Use the Freeform Pen to draw a rough outline around the object. (Make sure you have the "Paths" option at the top left of the screen selected.) Generally err on the side of cutting into the image rather than leaving a gap.Use Ctrl+Click to manipulate the points, and drag their "arms" around to change the curve. To insert a new point, just click on the line.
When you've finished, right-click the line, and click Make Selection. This turns your line into a selection.
3 - Removing the background
Cut out your object and paste it into a new layer. (Ctrl+C, then Ctrl+V) You can then replace your background layer with whatever you want.I've put in this magenta (opposite of green) background to show up any green bits that might still be stuck to our budgie.
Notice there are no jaggy edges, and there is no visible green outline.
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