Word for the inability to explain something to someone due to their complete lack of understanding

That person is "out of their depth", but you are "at a loss".

Both from the free dictionary:

be out of your depth - to not have the knowledge, experience, or skills to deal with a particular subject or situation; When Ruth started talking about the differences between the databases, I knew I was out of my depth. By half-time, England was losing 4-0 and the English players were looking hopelessly out of their depth.

Adj. 1. at a loss - filled with bewilderment; "at a loss to understand those remarks"; "puzzled that she left without saying goodbye"

In a less formal setting, you might instead say that they were talking out of their ass, rendering you speechless.

From the Urban Dictionary:

talking out of your ass To make an obviously false claim. To attempt discussion on a topic with no prior knowledge of it usually results in this. The moon landing was a hoax? You're talking out of your ass.

or

When someone says something stupid that they have no proof of. For example, "The Space Shuttle was destroyed by TERRORISTS!"

From the Free Dictionary:

speech·less (spēch′lĭs) adj. 2. Temporarily unable to speak, as through astonishment.

Neither single words nor specific to your context, but the general situation you describe seems similar to the “hopeless endeavor” and “exercise in futility" that Vika encountered and eventually abandoned when debating religion in Katya Cohen’s book ‘The American Spellbound’ found on ‘Google Books.’

As for your example, I would simply say that the statement is false (you're correct in noticing that not everything that is false is also necessarily nonsense). Amusingly enough, according to Wikipedia this can be further pinpointed as a "(false) statement made out of ignorance". But the best single word to describe what you're talking about, AFAIK, is just "false".

I don't think there is a word or an idiom for the situation you are in, but the person who wrote that could be said to be out of their depth:

Fig. involved in something that is beyond one's capabilities.