Dead easy!
Download, install and open GIMP:
http://www.gimp.org/
Don't worry if it takes several minutes to load; that's not unusual!
Go to File > Open.
Navigate to the location of your image file and double-click on it.
Drag the edges of the image window, so that it more or less fills your screen (while still leaving the toolbox on the left visible).
Go to View > Zoom > Fit Image in Window.
Decide what colour you want the new background area to be. It needs to be shown in the solid block of colour in the toolbox on the left. If it's just black or white then you can switch between the two simply by clicking on the bent double-ended arrow at the top-right of that block of colour. Otherwise (probably better), to choose a nice colour, click on the block of colour itself. Then click on the vertical 'rainbow' band for the approximate colour you want and click again in the big coloured square for the exact colour.
Click on the 'free select tool' in the toolbox. (It's the one with a lasso)
Now the tricky bit: Holding the left mouse button down, carefully draw around the edge of the 'wanted' bit of your picture. When you get back to where you started, a mustard-coloured blob will appear, telling you that the program has recognised that you've completed a closed loop, so you can then release your mouse button.
Right-click inside the selected are and go to Select > Feather. Getting the correct value at this point can be a bit hit-and-miss but I suggest entering 10 pixels before clicking 'OK'.
Then right-click again and go to Select > Invert.
Go to Edit > Fill with FG Color.
Right-click and go to Select > None.
Optional: If you want to crop your image, click on the 'rectangle tool'. Click at the top-left of the are you want to crop and, holding the left mouse button down, drag to the bottom-right of the required area. Right-click and go to Image > Crop to Selection.
Go to File > Export As, and save your file under a new name.
Job done!