Why is “Ten Items or Less” wrong and “Ten Items or Fewer” correct?
“More” or “less” refers to an amount of a substance - “there’s less water in that glass.”
“More” or “fewer” refers to a quantity of things - “there are fewer ice cubes in this glass.”
Grammer people why is this wrong?
Because “grammar” doesn’t have an “e” in it, and you’re missing some punctuation. 
To add on to what Gary T said, if you can say “There’s two <whatever>,” and have it make sense, such as “There’s two ice cubes,” then fewer is appropriate. If it doesn’t make sense, such as “There’s two water,” then less is appropriate.
Of course, these days the distinction is relegated to more formal contexts, with less doing double duty in informal contexts. Most people don’t give a damn about grocery store signs, but you can bet it’ll be noticed in a published study.
“Fewer” goes with count nouns, and “less” with non-count nouns. Well, in general. However, it seems to me that the usage “ten items or less” is probably more natural sounding than “ten items or fewer” to most people.
However, as with many grammatical issues, there is some controversy surrounding this, especially because it does sound a bit pedantic and stilted to many speakers of English.
Here’s an excerpt (WARNING: PDF) from Merriam-Webster’s Concise Dictionary of English Usage.
I believe you mean “There’re two ice cubes.”
Well, sure, if you want to get all pedantic about it. 
Well, not necessarily, and if it is noticed, it won’t be because it stands out as awkward to English ears, but because some people have trained themselves, for whatever silly reason, to notice such things from time to time and get irrationally upset over them.
As pulykamell’s cite explains, what we have here is a feature of English that has been thoroughly ordinary for more than a millennium. About two hundred years ago, some guy came along and expressed his own idiosyncratic preferences, but this never really took on with the English speaking public at large, and thus never really became a true rule of English grammar, though it ended up being codified nonetheless in an awful lot of wrongheaded usage guides, the sort which rarely bother to take a glance at reality. (Another choice quote from the excerpt: “This approach is quite common in handbooks and schoolbooks; many pedagogues seem reluctant to share the often complicated facts about English with their students.”). Basically, this is the sort of “rule” which you could never hope to learn by observation of the actual speech of English speakers and their natural linguistic judgements; it only survives vestigially, as a kind of virus, self-reproducing in ill-founded proscription-tomes, misguided admonishments, and other sources of prescriptivist propaganda, in a ritual quite detached from the realities of the language.
Either is correct, depending on how you imagine the rest of the phrase:
Ten items or fewer [items].
Ten items or less [than ten items].
Arguably, the former is better as the omitted portion is directly parallel to the stated portion, but since it’s not a complete sentence as it stands, pick your favorite.
ETA: It occurs to me that someone may suggest that the second sentence should be “Ten items or fewer [than ten items],” but as a native speaker that sounds wrong, and I do make the less / fewer distinction in ordinary speech. I’m trying to figure out why, and the best that I can come up with is than a dependent clause fits better in the non-count category than in the count-category. Anyone have an explanation?
It should definitely be “fewer than ten items”. Just adding in the rest of the phrase that is understood doesn’t alter the countable versus non-countable distinction.
“Ten items” is countable, so it’s “fewer than ten items”, simple as that.
However, note that Bosstone is located in Arizona. Between 2:45 and 2:50 (your posting times), the ice cubes melted. Therefore, there’s a puddle of water is correct.
Unless the cubes were at some distance from each other, then there’re two puddles of water on the table would be right.
Perhaps, though, the cubes weren’t actually Bosstone’s to begin with, but belonged to his friends. Then they’re rather upset that there’s nothing left of their ice cubes would be appropriate.
Except that the cubes actually belonged to a New England poet, so Thayer’s soothsayer predicted there’d be no joy in Phoenix, as Bosstone, mighty Bosstone was advancing toward the pair…
I need help.
There’s no “dependent clause” around there; indeed, there’s no clause at all. We can see this by noting that a clause contains a subject and a predicate, while “ten items or fewer [than ten items]” does not even contain a verb.
As per pulykamell’s cite, “less than ten items” is perfectly grammatical as well, even though “items” is indeed countable.
Artificial, prescriptive, pedantic nonsense up with which I shall not put.
I don’t. If “ten items or fewer” sounds good to you, I don’t see why “ten items or fewer than ten items” sounds wrong to you. They both sound equally pedantic, but “correct” (as defined by the count vs. non-count distinction) to me.
If you want to go the ellipses explanation for “less than,” the usually argument is that it is interpreted as:
“Ten items or less [than that amount].”
“[Than ten items]” should still take “fewer” by the usual rule.
An aspect of “countableness” that is sometimes missed is that “fewer” is used when the things counted are objects, not continuous quantities. Hence, “I have fewer apples than Joe,” but “It’s less than (not fewer than) ten miles from here.”
Although you can count miles, pounds, etc., when speaking of distance, weight, time, and other divisible quantities use “less” instead of “fewer.”
I get to be the first to point out that it’s not a grammar question, it’s a usage question. Either form is equally correct gramatically.
Think about how many of us learned math.
Five is less than 10. A < B. Etc.
It is not such a leap to go from that to "ten items or less’’.
I feel stupid now about calling it a “dependent clause.” Oh well.
I’ve always assumed the less / fewer thing was a distinction I made natively, but I wonder now if I didn’t just learn that rule as a hypercorrection some point in childhood, and therefore don’t apply it consistently.
It’s painful, but I do love the way this board takes me down a peg and challenges my assumptions about what I think I know.
Right on. It’s a style thing.