Why the language change?

I too have wondered about this when I watch the show. It seems strange to start with subtitles and then suddenly switch to English in this manner. At first I imagined maybe they wanted to do subtitles originally but then they ran out of subtitle budget really early on and had to switch. This of course doesn't make much sense.

I don't think there is anything more to it than to have a creative way to show the viewer that while the language being heard is English, the characters are in fact speaking Russian. By having them start speaking Russian it establishes the fact that there is a different language, but then there is a transition from Russian to English, including a visual transition to help illustrate the effect. It is a creative way to do this.

As for the point of transition, it is noted in the IMDb Trivia section for the movie that the word "Armageddon" is the same in English and Russian although pronounced differently. This is likely the purpose of choosing that word as the point when the character's transition from Russian to English.

Like @sanpaco said, armageddon is the same in both languages. IMDb Link

My understanding was simply that it would be easier for audiences to follow what was happening. However, it could be that the director simply liked the idea based on a movie he had seen: Reddit Link

Patrick (H) Willems argues that the language change is a way to indicate that the characters who we assume to be the "bad guys" (the Russian submarine crew) and who have been kept distanced from the audience by speaking Russian, are in reality "good guys", and the language switch is meant to have us empathize with them. The actual villain of the movie is the word "Armageddon". The subtitles required to understand the Russian characters act as a barrier to the (intended) audience which is mainly English speaking.

Willems starts off his argument this way:

If you're watching an American movie, that means it was made for an english-speaking audience. And outside of very few exceptions the main characters will speak English. In the occasions when characters are speaking the language other than English and the dialogue is subtitled, audience members have to exert the tiniest bit more effort to understand them. As Bong Joon-ho put it in one of his many Oscar acceptance speeches: subtitles are a one-inch barrier; it's so small, barely an inconvenience, but it is still a little barrier that does not exist with English dialogue.

So when you're watching an American movie in which multiple languages are spoken, this has a small subconscious effect on us. We will naturally connect more to the characters speaking our language since the film was put in the other characters at a slight distance via that language barrier.

He points out that McTieran uses the same trick later in the movie as well:

All this time you know the movie is built into a scene where captain Ramius and the soviets are gonna meet Jack Ryan and the Americans. Our two main characters have to come face to face. Now this whole time we've been hearing them both speaking English but really we know the Soviets are speaking Russian. So what do you do here, how do we show them talking to each other? The key thing McTiernan does here is focus on perspective: the language we hear depends on whose perspective we're in at the time.

The scene begins with the Americans. We follow them boarding the Soviet submarine and from their perspective the Soviets are speaking Russian. This time though there are no subtitles: the Americans don't speak the language so the audience doesn't understand it either. Suddenly this ratchets up the tension. We want these people to understand each other and work out an agreement, but now they can't communicate. We see the Soviets as the Americans do as potential enemies who can be saying literally anything -- classic Tower of Babel scenario.

The first shift in communication is nonverbal: Jack Ryan gestures to a Soviet asking for a cigarette. It's just a physical gesture but it's something universal that they all understand. And as Ryan takes the cigarette and this small bond is formed, now the Russian dialogue becomes subtitled. It's my three is your junior duck. It's a step forward in understanding. The playing field isn't even yet but it's getting there.

And then McTiernan uses a similar trick as in the earlier scene: Ramius says a line to Barden in Russian but the last word in the line is "kuru", another word that's the same in English and Russian. And at this word Jack Ryan laughs in this scene where you have a tense standoff between opposing militaries from opposite sides of the earth who can't communicate. The moment in which they connect and empathize with each other comes via word they both share.

And here we learned that Jack Ryan can speak a bit of Russian. For a couple lines he and Ramius speak via subtitled Russian, and then they both shift to English. The film lets us know that they're still really speaking Russian since commander Mancuso clearly can't understand. But now our two protagonists are finally on the same level: they're speaking the same language as the audience. Contact has been established; the language barrier no longer matters