틀리다 means to be wrong, and 모르다 to not know or to be uninformed. These are verbs making up the predicate of a sentence in their simplest form (e.g. 나는 그 사람을 모른다). 틀리는건 당연해요 uses the concept of 틀리다 as a noun, like "that (you) are wrong" or "being wrong". So you need a construct to turn the verb into a noun, and the "-(ㄴ/은/는/를) 것" is the most common way to do it. The ㄴ/은/는/를 ending of the verb turns it into an connective form like an adjective with a specific connotation: the verb has either completed(모른), is going on now(모르는), or is likely to happen(모를). And 것 after it captures it as a noun phrase. It is similar to "that you do not know" in English, for example. So, - 틀리는 건 당연해요 = [틀리는 것]은 {당연해요} = {It is only natural} [that you get it wrong}. - 모르는 건 당연해요 = [모르는 것]은 {당연해요} = {It is only fair} [that you don't know it].

It is a abbreviation of 거는 (것은 -> 거는 -> 건) Here it is a part of the noun forms of the verbs 틀리다 and 모르다. And the ㄴ simply means it has the emphasis of "topic".