A more important question is "why do people come to the boards and post here?" The vast majority of League players don't and never will. Some players do come here and post, and that makes them different. That by itself is enough to say the group isn't representative. On that alone, I can say that taking the boards to represent the general League population is statistically invalid. People come here for quite a few reasons, and assuming those reasons are shared among the League population in general is invalid.
However, your question is important, and it does have an answer. When people go on the Boards, what do they see? Probably something like "Zed sucks" "Dynamic Queue sucks" "Fizz sucks", etc (I'm not taking any sides on these issues, I'm just using them as examples). These show up all the time, and almost more importantly, opinions disagreeing with them are almost universally downvoted and ridiculed. This means that the forum promotes a very homogenous view, and they are hostile to any views that are not the majority here.
Basically, if I actually thought that Dynamic Queue was a good thing or that Fizz was a well designed champion (and again, I'm just using these as examples), I'd maybe post here once before I got shouted down and insulted. If I said that I hated dynamic queue and that Fizz needs to be nerfed into oblivion, I'd get praise and people would agree with me. That encourages more posting.
The boards are designed to only show one opinion. They are not favorable to representing the beliefs of everyone in League. They represent the members of the Boards. Nothing more. When the guy I responded to tried to use statistics to say that the boards represented the League population, he was dead wrong.
tl;dr: This is nothing more than sampling bias. The entire argument and conclusion is invalid because he has selected a nonrepresentative sample.