It's a construction that doesn't translate word-for-word to English (might count as idiomatic). When I took German in school, we learned the "<subject> <verb> gern" as the way to say "<subject> likes <verb-ing>", e.g. "Ich wandere gern" for "I like hiking", and only used modal verb constructions later. That being said, I wouldn't say "I meet new people gladly" should not be accepted, since although it sounds odd in English, well, so do many DL sentences....
jemanden treffen = "to meet someone (by chance, unplanned)", e.g. to serendipitously run into someone you know while shopping
sich mit jemandem treffen = "to meet [with] someone (planned)", e.g. to agree with a friend to have coffee together one afternoon next week
Using pons.com will help with this kind of thing.
I think it is indeed because there's no mit in the German sentence. Even in English, "to meet with someone" implies that it was planned in advance but "to meet someone" doesn't necessarily. These roughly correspond to the difference between sich mit jemandem treffen and jemanden treffen in German, although the latter (as in this translation exercise) is usually explicitly unplanned.
They are interchangeable. But using using one might just sound better than the other. Gern is just a shortened version of gerne, since sometimes the -e on the end of words is dropped off in casual usage.
Duo gets picky at times and on others a looser translation is accepted. I totally understand many things in German, but sometimes cannot find a translation that Duo likes. As your comment suggests, you know that language does not always have an exact translation. Thank you for your comment.
This little sentence has caused a flood of comments. Basically, I would not blink if a German speaker spoke that sentence. I got it. The speaker likes to meet new people. If you want you can translate it using the progressive, since German does not have one, but the concept is the important thing.