You have to use what is called "feathering." This allows you to blend a designated amount of pixels in your selection with what the selection borders on. I'm not sure if this is available in Elements or not.
Yup, feathering is the way to go. I happen to have Elements 7 here and just verified that it does feathering (I have an older version at home and am pretty sure it's on there too).
In case you're not familiar with feathering, it's a refinement of the border when you select an area (using a selection box, lasso tool, whatever). Feathering creates a buffer zone within which whatever action you're performing is transitioned from full effect to nothing. In this case, assuming you selected the sky around the rocks, what you would get is that the sky right up against the rocks would appear as in the original and transition to your final version. Depending on what effects were used and how large a feather you select you may get a halo but you won't get a sharp outline like in this version. You can also, rather than simply entering a value in the feather box, choose to "refine edge". This will not only let you feather but expand/contract the selection area and smooth corners as desired. Expanding/contracting the area in combination with feathering will change where your transition falls (for example, maybe it moves any halo slightly onto the rocks, where it may be less visible). Can't really give you guidelines on what values to use as for me it's still trial and error (or as I prefer to call it, iteration) based on the individual image and effect being used.
The bummer for you and this image is that feathering is done as part of the process rather than its own step, so you'd have to go back and redo what you did before with a feathered edge. If you don't think you can recreate those steps you could try blurring or otherwise attacking the edge on this image, but I think any of those methods are likely to soften the outline of the rocks. Feathering transitions the
effect but would keep edges sharp.