Open the flower image.
Click on the Paint brush tool. Turn on the mask mode (Lower right side of screen. Change color to white. For this image, accept the standard default values for Soft Edge, etc. You will paint out the mask with white paint. Adjust brush size as needed. It's a good idea to work in magnified mode. Start painting over the part of the image you want to KEEP.
Increase brush size to paint out large areas, decrease it for smaller areas. Don't worry if you paint out too much. You can paint it back in using black paint. In the example below, you can see where the paint went out over the edges. Changing to black paint allowes one to paint the mask back into the image.
When you've painted out the entire flower, click on the mask to turn it off. There will be a selection line around the outer edge of the flower. If there are selection lines inside the flower (see example below), Click on the Paint Brush again, color black, then turn the mask back on and paint out the missed area.
When you're happy with your image, turn the mask off, right-click on the image and select Convert to Object from the popup menu. Drag the image to a new window. It will have a nice clean edge. To verify this, fill the background with a dark color paint. You won't see any white jaggies here! Drag your new object to the Easy Palette if you wish to keep it or simply save as a .ufo file.
Note: When working with really small areas, reduce the soft edge to somewhere around 25.