While I haven't read 'The Travels of Marco Polo' in quite some time, I don't recall a specific passage where he talks about which languages he knew; he only says he knows four (outside of his native language). Which four is up for speculation. There's a good chance he would have learned French, and likely Arabic too. That leaves two, which could very easily have been Mongolian. Considering that Polo spent more than a decade in the service of the Mongolians (if we are to believe his tales) learning Mongolian seems very likely.
That said, how they would have communicated at the time of their first introduction is probably through a translator, likely one that spoke both Arabic or Persian and Mongolian. Polo's father had spent considerable time on the silk road and would have had the time (and the need) to pick up at least some of one or both of those languages. It is also possible that the father learned some Mongolian as well and taught his son prior to their arrival.
However, fluency that the characters demonstrate is something else all together. They make a point of demonstrating that language differences matter in the first episode, and then essentially ignore it for the rest of the show, so what I take from that is plot contrivance more than anything else.
Not a historian, but I think in the first episode it shows Marco learning Mongolian on his two/three year travels on the silk road with his father. I think it showed him learning from their guide, and I think he even says so in the narrative.
hi! other input on this is welcome, but fyi, there was an earlier post that may be of interest