There are two factors for making connections through conversation. You've pretty much nailed the "showing interest" part.
But don't forget, in addition to needing to show that you are interested, you need to show that you are interesting, too.
A good conversation will have some back and forth between both sides, on both factors. So merely showing interest may sometimes be counterproductive if you can't tie that intense interest into something to reveal about an interesting side of yourself.
For example:
Statement: "I went to France last summer."
Good response: "Ooh, that's really cool, what were your favorite things you did?"
Better response: "Ooh, that's really cool, I've always wanted to go. What recommendations do you have for places and things that I absolutely must experience?"
Similarly, tying things back to your own experiences may be helpful at generating some of that back and forth, even on topics that you don't have any direct knowledge or experience with. For example:
Statement: "I was hesitant about the escargot/snails at first, and there just wasn't a ton of substance to them, but it was delicious under all that butter anyway."
Good response: "Wow that sounds like a fun experience, would you do it again?"
Better response: "Wow that sounds fun, so was it like linguine and clams where it's as much about the flavors that get infused into the butter as it is about the clam itself?"
I know too many people who get stuck in a pure listening mode with strangers and forget to actually show their own personalities and why they're worth getting to know, too.
The original screenshot is from Its Always Sunny, where the illiterate character has trouble pretending to be a philanthropist, because he can't pronounce the word "philanthropist," much less describe and understand what it means.
The comment you're replying to is just riffing in their own way, completely unrelated to the scene where the screenshot comes from.