Negative Polarity Items (NPIs) in English
Every language has lexical items which are restricted, seemingly arbitrarily, to specific contexts. Let's start with some English sentences to introduce the concept:
1a. I don't like her at all.
1b. *I like her at all.
The first sentence is fine, but the second sentence doesn't work because at all is a Negative Polarity Item (NPI)—a lexical item restricted to negative contexts.
Sometimes we have pairs of positive and negative polarity items:
2a. I like her, too.
2b. I don't like her, either.
Here, too and either are a positive-negative pair. If we swap them, the sentences don't work (at least, not with the same meaning):
3a. *I like her, either.
3b. *I don't like her, too.
Many NPIs can also appear in interrogative contexts—that is to say, in questions like 4c:
4a. I don't like her at all.
4b. *I like her at all.
4c. Do you like her at all?
But every polarity item has its own requirements, which can be arbitrary, complex, and context-dependent. For example, till is an NPI in clauses with a punctual interpretation:
5a. We won't leave till six o'clock.
5b. ??We will leave till six o'clock.
In 5a we interpret leave as a punctual event, one that takes place in a single point in time, but till can't appear in positive punctual predicates, so we can't interpret 5b the same way. Instead, we interpret 5b as durative—it sounds like we're planning on leaving for a while, possibly for hours at a time. This is, of course, nonsense, so 5b is highly questionable.
English is full of examples like these, and unsurprisingly, so is Japanese.
Negative Polarity Items (NPIs) in Japanese
Japanese has a number of NPIs. One of the first ones taught to learners is しか, which is always followed by some form of negation:
There's no reason in principle why this is true. Positive しか could make sense, and indeed 日本国語大辞典 notes that there are rare historical examples of しか unaccompanied by negation. But in Japanese as it's spoken today, the negative requirement is more or less an ironclad rule.
Unlike English, Japanese is verb-final, and negation usually comes toward the end of a sentence. Many Japanese NPIs appear early, and they're often signals that a negative predicate is coming up:
In many cases, this signal is strong enough that you can leave out the actual part of the predicate containing the negation (as long as it can be inferred from context):
This is true even for items like 全然 which have an association with negatives rather than a strict requirement:
(Of course, I haven't supplied any context here; you'll have to imagine the context that allows the verb to be left out if you want 8b and 9b to make sense.)
Your example too is among the Japanese NPIs:
There's no reason in principle why 何も shouldn't be possible in positive contexts. It would make sense, just like certain other combinations of question words plus も. For example, どれも is fine either way:
So is いつも:
So is 誰も:
Notice here the arbitrary and quite strong preference for が in 13a. Notice too the arbitrary difference between 知っている and 知らない, using 〜ている only in the positive. Every word has its own story to tell, and its own way of being used.
And 何も has its own story, too. Unlike these other question words in examples 11-13, 何も happens to have an arbitrary restriction to the negative, while 何でも does not—and you'll just have to memorize this difference.
To convince you that it's truly arbitrary, consider あまり／あんまり. The following examples are borrowed from a paper by Ai Matsui:
In this particular example, it seems that あんまり requires a negative context. However, this is not always the case. In the following example, it's licensed in a positive context by the presence of から:
Why? No reason. That's just how the language is.
Sometimes we can speculate as to why. Sometimes we can look at the historical evidence and try to find a reason from etymology. For example, some linguists suspect that しか contains a contracted version of the particle は, often associated with negative predicates. And it's true that we have relatively few cites for しか in positive contexts, even historically.
But generally speaking, why is a difficult and possibly unanswerable question when it comes to polarity items. For us learners, they're just something we'll have to memorize.
In this answer, the * symbol marks a sentence as ungrammatical.