A. “Shechita” literally translates as Kosher/ritual slaughter, but can also cover the entire Kosher meat/poultry preparation process. I can think of 3 reasons for preference or insistence on specific Shechitas (you may see “Chassidishe Shechita” or “Lubavitcher Shechita” or “Shechitas Beis Yosef” which some Sephardic Jews prefer), some of these reasons may or may not apply with the person or situation you are referring to.
(1) Kosher is Kosher, but there are additional stringencies or concerns that “super-kosher” Shechitas are careful about. For example, maybe the Shochet checks his knife more often than in the standard OU or OK operations.
(2) In addition to the skill and training of the shochet (ritual slaughterer) there’s also a requirement that they be G-d-fearing and (piously) observant, which is another reason some people may prefer one of their community, or a community with their level of observance) to be the Shochet. In fact, back in Europe, people were sometimes more concerned with the religiosity of the Shochet more than that of their Rabbi.
(3) It can sometimes be simple economics, akin to the “buy local” campaigns. If you buy meat or chicken that is certified by your community, you keep the money closer to home. This may not be the reason for everyone, but it may play some role.