Q: What are the core skills for a producer?
A: From my experience, the core skills for a producer vary but generally include: people management, negotiation/mediation, coaching/mentoring, solid knowledge of game development (including the area which they oversee), contract review, time management/scheduling, presentation/public speaking, and mastery of software tools for the creation of spreadsheets/slide decks/documents and so forth (i.e. MS Office or similar). Specialized knowledge of production techniques can come later--having several of these will get you in the door at least, regardless of your background.
Q: How do I develop the skills needed to become a producer?
A: The best way to develop any skill is by using it. If that's not an option (i.e. because you aren't in the position yet), you should be able to read books, take classes or find colleagues who can help you in person. There are also lots of skills that aren't specific to games--public speaking and presenting, for example, can be learned/enhanced through Toastmasters International. You may wish to take online project management courses if you really want to delve deep into that specialty. If you just want an overview, I highly recommend the book The Business Savvy Project Manager by Gary Heerkens. It is by far the best overview of indispensable project management skills for anyone, minus all of the stupid bullshit that isn't really useful.
Q: What are employers looking for in an entry-level producer candidate?
A: Employers are looking for someone who fits the job description completely or partially. If partially, enthusiasm, pragmatism, intelligence, and diligence can make up the deficit. If you are seeking an entry level position, you'll have to differentiate your application. I suggest finding a professional resume template online and using that to draft a polished resume. Make sure that you have ZERO typos or grammatical errors in your resume and cover letter. Be sure the correct company name and job title appears in all correspondence (believe it or not, I've seen plenty of applications with cut-and-paste errors).
Most importantly for anyone trying to "break in" from another area or even industry: find ways to take what you've done before--game-related or not--and show how the skills you gained or problems you solved can be applicable to the job to which you are applying.
For example, if you were in QA, explain that your daily responsibility was to evaluate the state of the release and to bring critical issues to the attention of your boss/lead. Describe the tools you used (showing technical savvy & familiarity with engineering process) as well as communication channels (email, discussion, meetings, showing that you are a team player and understand group dynamics). The best way to make this list, in my opinion, is to write down each task you do and the break it into components--did it require tech, communication, team work, specialized knowledge, problem solving skills, etc.? This way you can show an employer that you have skills that are applicable to their needs, even if you don't have direct job experience.