I did study Spanish in high school, so it's hard for me to judge as far as Spanish is concerned. However, when I spent a summer semester in Barcelona, some Catalans adamantly spoke only Catalan. If I spoke slowly they could understand me and I could understand them. As Duarte pointed out, an occasional word would have to be explained in a sentence.
As for Portuguese, I had a wonderful Portuguese nanny for a couple of years. She became like a dear friend to me and remains so. Again, I spoke in Italian to her and she spoke Portuguese to me, and we got along just fine. My friends were often quite amused to hear my son jabbering away in Portuguese. Unfortunately, he's lost most of it. :)
It may be my imagination, but I thought perhaps my ease with both Catalan and Portuguese had something to do with the fact that I grew up hearing the Ligurian dialect. It's closer, imo, to these languages than is "standard" Italian, i.e. Tuscan. Or maybe it's just that since the "core" of the word is usually what is spoken in those dialects, I recognize it in other languages.
Or perhaps it's as one of the people on the video said: for some reason Italian speakers can understand other Romance languages better than they can understand us.
As for French, I found written french very easy. (I read somewhere we have more cognates with French than other romance languages.) Pronunciation was a little harder.
Then, I think there's a difference person to person. Some people just can make the "leap" from language to language faster than others.
Maybe you could try seeing how this sounds to you...
https://www.youtube.com/watch?v=k1RaZx8qbLc
And this...
https://www.youtube.com/watch?v=CJqHV-lo0oE