Examples:
He got that left arm in there.
He almost made it to that down marker.
I fully understand that language is what people speak, and I am not a stickler for correct grammar, but this makes me itch.
It’s probably common in all sports broadcasting, but football is the only sport I watch. 
Your examples seem like perfectly normal, if a bit colloquial, speech. They certainly don’t seem like examples of a type of speech that’s unique to NFL announcers.
What would you consider ‘corrected’ versions of your examples?
He got his left arm in there.
He almost made it to the down marker.
Something like that?
Yeah, I think this is just mostly normal speach patterns that creap into all forms of communication. I don’t think it has anything to do with NFL or sports, it happens everywhere.
What does YA mean?
‘Yet Another’ would be my guess…? Probably not ‘Young Adult’.
The words “the”, “this”, “that”, “these”, and “those” are all closely related, originally being different case-endings of the same word. Vestiges of the evolution to contemporary usages are out there.
My take is that the announcer is describing something you’re both looking at, so it’s almost as if he’s pointing to it as he’s speaking. (He may even be doing exactly that with the telestrator.)
So it emphasizes that it wasn’t just any old left arm – it was that one, right there.
Related: Ever notice that when a defensive lineman deflects a pass or a kick, the announcer always says “he got that big ol’ paw up there”?
In the NFL it means Yelberton Abraham.
Yet Another NFL question.
And what I’m saying is that “his” would be so much more descriptive. “His big 'ol paw” make it clear that he didn’t grab someone else’s big 'ol paw, but quite definitely used his own!
Straight from the dictionary (emphasis added)
determiner
determiner: that; determiner: those
1.
used to identify a specific person or thing observed or heard by the speaker.
“look at that man there”
referring to the more distant of two things near to the speaker (the other, if specified, being identified by “this”).
2.
referring to a specific thing previously mentioned, known, or understood.
“he lived in Mysore at that time”
3.
used in singling out someone or something and ascribing a distinctive feature to them.
“I have always envied those people who make their own bread”
4.
referring to a specific person or thing assumed as understood or familiar to the person being addressed.
“where is that son of yours?”
Seems perfectly acceptable to me
By most (I have no statistical data to justify using “most”) people, “that” is used in the #1 sense, to refer to a specific one of similar objects. That ball (not some other ball) is the one he caught. There is only one ball in play at a time, there is only one first down line to gain.
Like I said, I’m not trying to be overly pedantic, but it grates when I hear it. He caught the ball. He made it to the line to gain.
Yelberton Abraham “Post Tittle Here”
I like it.