The question is: What is the purpose of the hand salutation at the beginning of Tekki 1 (Naihanchi 1)?
Here's a video showing the whole form:
One possible answer is: It's a bear hug defense.
There are other interpretations (bunkai) for this movement, but that one makes the most sense to me.
If you're new to kata bunkai, you should look at some of my other answers to explain what bunkai is in general, and why Shotokan might not teach it as it was taught in Okinawan karate:
Why is more time dedicated to exercises and very less for sparring? Is it for the fee?
Name and meaning of stance where you stand with fists on hips?
Styles of Karate
Getting back to the first position of Tekki 1:
How is this a bear hug defense?
Imagine someone is behind you giving you a bear hug. He has pinned your arms to your body. What do you do?
The first movement of the kata is the "salute" position where you have both hands open, palms facing down, one on top of the other. You're holding your hands out on a diagonal angle away from your body here. That's the first position.
In styles of karate that derive from Tomari-te lineages, such as Wado-ryu, you can see an extra flourish on this salute:
In that version, you'll see there's a lot more going on here. There's a big circle you complete with your arms. And you perform the hands together motion (the salute) not just low but high first before going low.
Furthermore, notice how his elbows go upwards in the next movement after that, shrugging as he's stepping out.
These movements imply this bear hug defense a lot more strongly than Tekki 1 does.
The motion you make with your hands pressing down in Tekki 1 is the end of a motion that's used to break the bear hug hold, or at least in order to free your arms from being pinned. It has been simplified in Tekki 1 to the point where it's completely hidden.
Imagine you're being held in a bear hug. Your arms are pinned tightly to your body. What do you do? Bring your hands together and slide them up your belly, under his grip. Breathe out. You're trying to move your hands upwards along your center line, wedging them under his arms. Once you get your hands above his arms, you can push down on his grip at his wrists.
By pushing your elbows outwards and upwards (shrugging), you are applying leverage against his hold, making it harder for him to maintain it.
At this point, your hands go back down. They will land on top of his wrists, your left hand on the top of his left wrist. Your right hand on top of his right wrist. Now, push straight down. It is designed to break his hold and open up his arms, thereby releasing you.
In some versions of this kata, your butt shoves backwards as you do this, and you lean slightly forwards. That is done to give you even more leverage against his bear hug grip.
I didn't describe what that big circle motion was at the beginning of Wado-ryu's kata version. It's not in most other styles of karate. I think it was added on to the tomari-te version only, but I could be wrong. And my opinion is that it is also concerned with bear hug defense, but clearly your arms aren't pinned here, because they can make that sweeping circle motion. That means it was probably done to prevent the bear hug in the first place, when you first start to feel him reaching around you. And then you grab onto his wrists and bring them down low. Either that or the bear hug was done without pinning your arms, and then you're going to take your hands and place them on top of his wrists, which gives you the same starting position as before.
So getting back to Tekki 1, that opening salute is really an encoded technique for dealing with a bear hug. You're pushing down on his wrists after getting your hands up and out of being pinned against your body. That will break his grip, opening up his arms. From there, you step out to the side. That frees you from the bear hug.
So you've used leverage to break his grip at its weakest point. Then you've created an opening. Then you're stepping through the opening to avoid his bear hug.
Now, Tekki and Naihanchi are all done on a line with the horse stance. That doesn't mean your applications will all be in a line just like that. Your opponent is not always to the side of you. But in order to get to an opponent, it's just assumed you will turn to face them and then do the technique that's shown in the kata. The kata isn't doing that for you. It was designed to teach, not to be a perfect representation of reality. It shows you the bare technique and nothing more. Then a good teacher would tell you how to interpret it and apply it in the real world.
And so when you step out to the side in the first step you take in the kata, the guy is still behind you. So you're going to pivot and turn around to face him as you do this.
Now you are facing him and have your right forearm checking (pressing against) his right forearm. That allows you to sense what he's doing and to prevent him from attacking you with his right arm. Your next technique is an elbow strike with your left arm, slapping it into your right hand.
Where does that elbow strike hit? Whenever you slap a strike into a hand, it typically means that the open hand is grabbing something and holding it there for you to hit with the other hand. In this case, you're reaching up to grab his head. Then you're slamming it with your elbow. Ouch!
Where you target on the head is up to you. Pretty much anywhere is going to hurt. Think about the nose, the temple, the jaw, or the base of the spine.
This is one of the most highly revered kata in shorin-ryu. It contains one nasty move after another. Almost everything here is designed for maximal impact. Bone breaking is a central theme. There are stories of karate masters who only practiced Naihanchi and nothing else, for decades. To them, these kata encapsulate the essence of karate.
Hope that helps.