People invent languages for all sorts of reasons. They have been made to exhibit artistic beauty, vivify fictional worlds, unite different peoples, investigate the structure of grammar, and implement philosophical visions. My reason for inventing languages is none of the above - I explore the essence of language for the same reason that I am interested in learning language.
Language is built right into our systems of perception. Through years of exposure, we learn to separate the sounds, structures, and perspectives of a language. We acquire this skill to such a high ability as to be able to instantly verify and correct mistakes in complex grammars. What if we applied this skill to other domains? I can only imagine what we might unlock by engraving such a strong ability in some other domain into our perceptive structures.
It is for these reasons that my favorite constructed languages are probably Lojban and Toki Pona. In diverging so far from the linguistic norm, I feel like they are the best in their class at poking at how we perceive information. I have also taken a lot of interest in learning braille, morse, and sign language. Doing so will require me to learn to perceive different physical ways of encoding information.
I want to go a lot further though. Although I haven't got the time to flesh out these ideas for constructed languages at the moment yet, I maintain this list anyways:
I think... I like playing with the modality of language, because I think language and board games and music and reality are all different reverberations of the same pattern.