Maybe don't assume that every little grammatical construction in English (or any other language) has a name. They don't, necessarily. And this doesn't appear to (yet). Usually, colloquialisms only "earn" names by being linguistically (and especially socio-economically) interesting not only to researchers, but also to the common people.
It is a relatively recent (several decades, maybe even over half a century, reflecting on movie dialogue) North American English colloquial expression. It's probably "jarring" to you because you're from an earlier generation where it never spread, and/or it's not yet saturated your dialect's region.
It's appropriate in colloquial (casual) conversation. If you meant "when" as in, "How do I use it?" then:
Here, the progressive tense communicates a different nuance to the verbs than a simple tense. It's a kind of "growing" (it's often accompanied by "more and more") or a "still processing"/"so far"/"as of this moment" meaning sometimes tinged with a hedging, non-confrontational, or cautious tone. It communicates the feelings of the present moment as they are in a state of transition (hence the progressive), and often a desire to hear more from the other party to the conversation.
So, you can use it when you want more/less information: "I'm liking it, I'm liking it," means, "Keep going; I like it so far, but tell me more."
Or when you propose something and want feedback: "I'm thinking we could catch the 3:15 and be in town for 5:00. (What do you think?/How do you feel about that?)"
You can use it when you want to communicate increasing emotional response to something (especially something constant or a daily event): "I'm liking my job more every day."/"I'm hating my job more with every passing day."/"I'm regretting my decision more all the time."
You can use it when you want to confirm your understanding in a non-assertive way, and get confirmation or clarification: "So, what I'm understanding is, you had an argument, you said something to him, and he ran away."/"What I'm getting here is that you two have some...baggage."/"I'm understanding/feeling that something big happened today, and you're not happy about it..."
Usually we use it in the present tense, but it can be used in the past tense too (though this can take on more of a pluperfect flavour): "Then, he was maybe thinking he could get away with it, that day. But he was wrong." Often, it becomes a pluperfect sentence, because we know what happened to that "growing" or uncertain element: "I was liking it right up until I noticed all the hidden add-ons and fees."