My son talks way too fast and often ends sentences with a mumble, especially when he is enthusiastic. It is really difficult to follow his conversation and I end up getting very stressed, after asking him over and over to slow down. The whole household seems to tense up when he's in full flow as we have to concentrate so hard to understand him. I'm conscious of making him feel bad or shattering his self confidence when I end up yelling at him to slow down or I angrily walk off. He is 12, gifted and talented at three subjects in school, really bright, and a gorgeous young man, well liked and popular. I've started to consider speech therapy and elocution lessons but neither seem quite right, I believe we need some professional intervention.
It wasn't clear from your question, but I'm going to assume in my answer that your son is able to communicate clearly when he makes a concerted effort, but perhaps makes it through a paragraph or two before slipping back.
My eight year-old has a frustrating habit of speech where he talks when no one is listening. What I do is simply not shield him from the consequences of his poor communication. If he enters a room and interrupts me properly (says my name and waits for eye contact before speaking), he receives the full benefit of my complete attention, and a less frustrated Dad who is more amenable to requests.
If he doesn't, I don't bend over backwards and frustrate myself trying to figure out what he said. I just say, "You know the right way to start a conversation. Try again later." He is still working on it, but has gotten much better.
Interestingly, I've never had to specifically define "later." He goes away, takes as much time as he feels he needs, then tries again, much more composed.
In your son's case, I would try saying something like, "I want to hear what you're saying, but I'm starting to get frustrated at not being able to understand. Please try again later."
I would also make sure you're holding up your own end. Pause TV or music, put your book or phone down, and give him eye contact and your full attention. If you can't at the moment, then tell him when you can.
Professional intervention sounds like a good idea -- at least an evaluation, so you could find out what's going on, and whether the professional thinks she could help him.
Your son's primary care doctor should be able to provide some suggestions. A speech and language specialist is one possibility. Occupational therapists are good at bringing fine motor and gross motor into balance with each other, and work on executive planning. They are a fount of ideas for fidget items that can have a calming effect (on you too, not just your child!).
My own experience with parental anxiety has been that it can be hard to let it go when it is no longer needed. To keep a lid on your own frustration and anxiety, just keep telling yourself that even though a 12yo feels an urgent need to be understood, it's not objectively as critical as it was when he was very small.
Here are a few (mostly untried) ideas for you to consider:
Have your son carry around a small notebook where he jots down some key words, a picture, an outline, the topic.
Keep a pad of paper handy in each of the places where he's likely to want to expound on something to you (car, kitchen, his bedroom, etc.). You can model the type of graphic assistance you need, by jotting down words, symbols and visuals while you're talking, and also while he's talking.
You could try a blackboard or a whiteboard -- see which visual aids work best in which situations.
Develop signals to show him when you are still with him, when you're tired and need a short break before he continues, when you didn't understand, when you're on overload, etc. It may take some practice to get him to look at you while he's talking so that he catches your signals (perhaps involving you touching his shoulder, leg or hand, or making some funny faces).
Take a sign language class together, and don't be afraid to make up visual signals for various things. The idea isn't to eliminate spoken communication, but to learn to augment the spoken word with visual gestures.
Learn to slow down your breathing when you're excited or anxious, and then teach him the same.
Try asking him to do a deep breath or two or three between sentences or paragraphs.
Music lessons in the instrument of choice can be very helpful in learning to control one's speed.
Build up his self-esteem: "You are so smart, sometimes I have trouble keeping up with you!" "I'm having trouble matching my speed to yours right now."
Perhaps it would help to have him do some aerobic exercise (like stationary bicycle, jumping jacks or jump rope, while he's talking, to force him to get out of breath!
There is a communication game I learned in high school involving three people, who take turns in three roles: A is going to talk to B about a topic of his choice; B does active listening; and C watches silently. The rules of the game are that A may only say one sentence at a time; B should repeat back the essence of what he heard after each sentence; if that was satisfactory for A, A should nod and go ahead, and otherwise provide a correction. If B isn't sure, he can say so, and A can try that sentence again. After a few minutes, you rotate into a different role. Perhaps you, your son and one other person could play this game together on a regular basis at a low-stress time in your day.
Play charades together. This could be a fun way to practice nonverbal communication and get in the groove of really looking at each other and noticing when the other person understands and when he doesn't, and when he's in the guessing role, he'll spend a few minutes or seconds in your shoes! Also, to play charades well, you need to plan how you're going to get your idea across.
Here's a really fun way to communicate that my spouse and I developed when one of our children was a baby and was having trouble concentrating on nursing when our voices got too excited. Pick an easy tune that you are both comfortable with, and fit your conversational words to that tune. We used "The Streets of Laredo" but you can pick any tune that works well for you.
I have a hearing impairment and have trouble understanding my 12yo if the acoustic conditions aren't optimal, or if I'm trying to concentrate on something else (e.g. how do I get to the music lesson from this part of town). It helps for me to indicate to my son that I'm not going to be able to take in what he's saying, at that moment. If he insists on continuing, I've been known to say, "Blah, blah, blah," which is something I can utter on autopilot, to show that I'm not going to try to understand, because it's all a wash of sound for me, but if he wants to think out loud, that's okay.