I have just been offered an opportunity to do ITIL certification. How will this benefit my career as a QA Tester? I have read about the course but I still dont get how important it will be for me.
Your title mentions ITIL training, while your question talks about ITIL certification.
Training and certification are not the same thing. You may find some benefit from the training, while the certification may provide little to no value.
To examine the training aspect, look at what you will learn during any classes, look at the syllabus, etc. Decide if these are things you want to learn, and consider if it will help you on your current career path.
To examine the certification aspect, look on job boards and see how many software testing jobs (if any) require ITIL Certification.
The following thread includes discussions regarding the value of certifications in the software testing profession in general: Do ISTQB/ISEB Testing Certificates prove that someone can test?
It doesn't answer your specific question regarding ITIL certification, but should be pretty informative.
The worth of the certification depends totally on your environment. It's definitely possible to do good quality assurance or testing without the certificate. On the otherhand, knowing more about IT services doesn't hurt, but if taking the course means that you can't participate in some other course, I think you can find better choices. Finally, when looking for a job it's up to the recruiter how much they value such certificates and up to you how much you want to work in an environment that would value the certificate.