The picture shows the mixer that is installed in the exhaust of a basic JT8D engine, those models up to a -17AR. As MD
-11 engineer pointed out, its purpose is to mix the hot, high velocity flow from the core of the engine with the cool, lower velocity fan flow to reduce the overall velocity of the exhaust gases. Since the primary component of jet noise is the shearing action of the high velocity exhaust gases against the ambient air, any reduction of the exhaust gas velocity will reduce the noise. Although it has been frequently stated on A.net that the mixers reduced thrust, a lot of testing was done to show that installation of the mixer did not alter the thrust-EPR relationship or cause a thrust loss. As ATLamt pointed out, hushkitted engines also have the inlet guide vane case spaced forward to eliminate the fan blades cutting through the wakes in back of the inlet guide vanes (kind of like how a siren works) that are the second loudest source of jet noise. Although it may not sound quieter to the ear, when measured on a gage, the installation of the hush kit did quiet the engine.
The picture shows the engine is installed on a 737, so there was no need for the tailpipe extension. Some of hushkits on DC-9s and 727's do have an extension, but it is not required on a 737. That is a standard 737 installation with the thrust reverser installed. This configuration also did not require the installation of the supplemental aft bracket. The supplemental aft bracket was developed for use on those JT8D engines that were going to have the Nordam developed 737 hushkit that included a suppressor and another mixer that were very heavy.