Importing CSV file to MySQL via PHPMyAdmin
What you're looking for is...

ESCAPED BY '"'


In CSV, one way of including literal quotation marks inside a single field is by doubling them up.

In the example file you posted, that's exactly what's happening at line 44.

The rest of your statement looks correct.  Replace the ESCAPED BY part with what I've shown above, and you should be good. 

http://dev.mysql.com/doc/refman/5.5/en/load-data.html


  Occurrences of the ENCLOSED BY character preceded by the ESCAPED BY
  character are interpreted as part of the current field value.


...which is exactly what you're needing.