Latin speaker here. This is a great question, and I am asked this question all the time.
I taught myself to speak Latin while I was in middle and high school (mainly eighth through eleventh grades). My main reason for doing so was that I enjoyed Latin as a written language so much (our classes were mainly reading and translating, with some reading out loud in Latin) that I wanted to create meaning in the language on my own, and that meant speaking, and later writing as well. I had no reason other than the desire to communicate and to master this new form of communication in all possible modes.
Communicate with whom, you may ask? Since the summer of 2015, when I attended my first Latin immersion program, I have conversed in Latin with well over one hundred people from six continents and at least sixteen countries. I speak Latin daily and use it to communicate with my Latin professors and grad students in Classics on a regular basis. (Granted, I chose my college because it has an active [=spoken and written] Latin program, so I self-selected this proximity of Latin speakers for at least undergrad.)
In the past few decades, there has been a "living Latin" movement among Latin instructors, including some professors in Europe (and a few in the US) and quite a few secondary teachers of Latin in the United States, that advocates speaking and writing Latin as well as reading literature and learning the grammar of the language. There are many different branches and different theoretical and practical approaches that make up this movement, but most of them share the goal of allowing the student to live fully in the language, such that the experience of reading great works of Latin literature is transparent and instantaneous, unmediated by translation or the grammar of their native language. See more info on this here (https://camws.org/meeting/2008/program/abstracts/11d1-5.Llewellyn.Soter.PatrickR.McGowan.Engelsing.html) and here (https://www.fluentin3months.com/speak-latin/).
For the first few years, I taught myself to speak Latin by constructing sentences out loud, which is hard. After that, I discovered (or was informed of) spoken Latin resources that came out of the living Latin movement in one way or another, and used resources like this video (https://www.youtube.com/watch?v=a61Dc_EFuI4&t=374s) to learn what spoken Latin sounded like. Being able to hear Latin gave me the mental model I needed to speak Latin conversationally. Even though I began speaking Latin on my own, I gained a great deal from the previous efforts of modern speakers of Latin, and eventually became part of a community of Latin speakers.
I speak Latin because I like speaking Latin. That is the simplest reason, and one that I can simply state, and not explain. I am of the view that, if I am going to learn a human language, I need to be able to speak and write at least some things in that language; otherwise, that language is not reaching its full potential in my use of it. (I am still working on this with ancient Greek; the effort to learn to speak ancient Greek is harder, since there are fewer speakers of ancient Greek than there are of Latin, and not as many resources either, although the latter condition is changing.)
I speak Latin to communicate and converse with other people, now, in this century. I speak Latin regularly with my professors and with friends who study or teach Latin. In the past month, I have discussed college sports, research design, and current events with other people in Latin.
I speak (and write) Latin to immerse myself fully in the idiom of the language, such that I can pick up a Latin text from 20 years ago, 200 years ago, or 2000 years ago, and read each with similar ease to how I read English. I can read history, mythology, ethnography, oratory, scientific texts, and breathtaking poetry in Latin; the effort (going on ten years) is absolutely worth it.
I speak Latin because doing so is great mental exercise, and forces me to think and craft the expression of my thoughts in a different cultural and conceptual paradigm. Latin is so much more than the predictable grammar rules I first fell in love with; many people like Latin for the "puzzle-like" challenge it offers, but I have come to love experiencing Latin as a medium of communication and human connection.