Besides manually checking the KWs on Google to see if your competitor sites show up, is there any way to quickly find out what KWs are giving most traffic to your competitor sites?
Yes, there is. Semrush does exactly what you are looking for. Grab a free trial (search on google for semrush free trial) and try it out for yourself.
My suggestion is Semrush. You can see competitor's keywords. Discover new organic competitors. Find the right keyword for SEO and PPC campaigns.
I too suggest SEMrush,it is used to get, Keyword Research, Competitor Research, Site Audit, Backlink Analysis etc.,
The best tools are *Semrush - You can know about the monthly search the keyword is getting and the CPC as well. By looking semrush chart you can check whether the keyword is profitable or not. *Open site explorer - Here you can check for the article which has best Page Authority compared to others and has more number of inbound links and root domains. *SpyFu - It not only show the number of organic keywords that the site ranks for but also shows how many organic clicks it gets.
K-meta.com - freemium tool witch help track on your competition's strategies i met it here at forum - http://blackhatworld.com/seo/free-simple-keyword-tool-just-try-it.764099/