What is the reasoning behind the Rabbi’s view?

A person is supposed to pray the way he speaks. By the same token, you could ask why speak with Sephardi pronunciation and not preserve the Ashkenazi pronunciation (or, for that matter, why speak Hebrew and not Yiddish). As far as I recall, Rabbi Kook and Rabbi Uziel disagreed about this. See Rabbi Kook’s approbation to the responsa Mishpetei Uziel.

“A person is supposed to pray the way he speaks” — meaning, all the more so than speaking before a king, or is there some other reason?

I didn’t understand what a fortiori argument you mean. Prayer is speech, and therefore one says it in the language and manner in which one generally speaks.