I have to agree with WallShadow; in order to subvert a system, you don't have to know the inner workings. Knowing only what SQL injection is, will help you attack a web-application vulnerable to it. But when you can use the SQL injection to get more information about the server (for example) and you determine that the OS running is not patched properly and thus vulnerable to a certain exploit, you have an additional vector for attack. You wouldn't have known that if you hadn't had the knowledge of exploits or weaknesses of the underlying system. So to answer your question: yes, knowing more about how systems work, will help understanding and can help you where you would otherwise fail. However: you don't necessarily need that knowledge to be able to attack. An SQL injection in itself can subvert a system, without having to worry about the underlying OS.
For me it works best if I know how the underlying system gets a webpage on my machine and the bits and pieces working together to make that happen. But that can be overwhelming (Apache, servers, ports, DNS, URL, IP addresses, routing, packets, POST/GET requests, etc.), that's why you should start slowly. Judging by your reaction, you already know what you want as an answer ("Should I really be...")
But then again, you should give yourself time to learn parts before getting to the bigger picture, just as WallShadow said.