Your interpolation (using IDW) is working correctly. The extent it has used is indeed the points. The points have a discrete set of x,y coordinates, they are not going to be polygons with several vertices (like large circles seen on the maps)
Those points in your image are represented by some sort of magnitude (probably representing the weighting used in the interpolation) giving the impression that they are larger than they should be.
They are not polygons, so the extent to be extended in this manner would produce further misleading results, as this is a prediction of values in areas where known values are not present.
When running the IDW tool, look in the environment settings see if you can set the Raster Analysis mask to that of the polygon in the picture (that's the extent you want right?).
But notice, that the extent outside the points will have results that are even more misleading, they give that rocket shape, like a plume because there are no nearby points to help distribute the weighting in the interpolation.
It takes that points value, and shoots it out until it reaches another point to enforce the interpolation (usually the p=0-2, also known as the Power), or less change as it goes down a path of least resistance
Adding other points, even with a value of zero, can produce misleading results.
But since this is a carto forum, accuracy in the information
(not the data
) may not be of importance
Data = what you see on the map
Information = ie; the attribute table
At the same time though, if you are trying to fill in missing areas
, there are none as the tool has executed properly.
Depending on the information being used, check out ESRI's IDW
help and look under the Usage section to see if the density of the points you are using meets some of the recommended criteria set out.