I won't go into details about the linguisitcs (experts in that domain will do a far better job), but just consider my own experience.
I'm French, and I've been living in Barcelona for 10 years. When I arrived here, I knew nothing about Catalan, but I had close to no problem understanding it when written, thanks to the knowledge of both languages (native in Fench and intermediate in Spanish). Some other latin friends of mine (Italian or Portuguese speakers) share more or less the same experience. It's harder for non-latin language speakers.
On the other hand, understanding Catalan when spoken is a totally different thing, and knowing back then French and Spanish was not of much help.
I've not been to any Catalan class, but after 10 years, you get pretty much used to it. I understand it without any issue, and I kind of manage to speak it (but I never found myself in a situation where it was a necessity).
In a nutshell, Catalan and Spanish are totally different languages, specially orally. However, they are close enough for a Spanish/French/Italian/Portuguese speaker to understand without much trouble.