The problem is, it's not a string. That is, there's additional structure. The quotes apparently mean something special, including that you shouldn't split on commas within them. (This quickly raises the question of what it means if there are quotes within quotes, and how you escape quoting, and...soon you're having to work far too hard.) (This is a common situation of course; I sort of wrote about it as if I hadn't seen it before, but it's actually a staple of programming.)
You don't tell us what the actual format is; GrandFather's guess that it's a CSV seems reasonable, but it's just a guess. So, if you're lucky, there's a clear description of what the rules for this CSV format are, and the input you get will reliably follow them, and you can write something to parse that. But the rule here is that parsing is hard -- if you do anything more than the most brute-force simple kind.
If you're really lucky, your input conforms to the standard that the CSV module that was suggested supports, and you don't have to write the parsing yourself.