I've been doing Pilates for over a decade now, and I've felt so many emotions. Sometimes in the midst of a session I feel exhausted and other times I feel almost overcome with pride at how far I've come. At the end of a session though I always feel happy and accomplished. The emotions of Pilates are unique to Pilates (at least for me, but I've heard lots of folks espouse the feeling of elation afterward). I chalk it up to the incorporation of Pilates principles that make the mental focus a key part of the practice.
The Pilates principles of breathing, concentration, control, centering, precision, and flow (whole body movement and relaxation are more recent principles) when cued throughout the session all help your mind relate to what your body needs to do. If you are cued to control your movement your mind goes to work visualizing and exerting more effort than if you were daydreaming while your body flails around.
This sets Pilates apart from other fitness approaches and even a lot of life activities. How often these days do you actively focus on the task that your body is doing? Maybe exercising this connection in a session is what leads to that commonly experienced feeling of, "ah, that was just what I needed to have a good day."