Anyways is in increasingly common usage, but is still considered to be incorrect. It is a colloquial or slang form of the word anyway - which is the word you have looked up.
Of the definitions in your dictionary link, the first is the meaning of the word in the sentence:
- (conjunctive) Regardless; anyhow.
The overall sentence carries a further shade of meaning as outlined below.
'Anyway' needs to relate to something, and as the title of the article doesn't have any context there is nothing for 'anyway' to relate to. Because of that, as a stand alone sentence the title does not appear to make sense.
However this kind of construct is commonly used. If you presuppose that the reader has an awareness of the subject (quartz crystals), but may not have detailed knowledge, then the use of 'anyway' is used to relate to that awareness which is the implied context of the title.
Against the background of this interest the question
What exactly is a quartz crystal, anyways?
is phrased in a conversational style to probe whether the presupposed awareness of quartz crystals is based on knowledge.
In the article itself, in the first paragraph it states "... would you ever have thought that quartz crystals would become pervasive electronic components ..." carrying on the conversational style and going on to provide information about quartz crystals and so anwers the question in the title. Interestingly, although the title does include the word exactly, it is only providing information into the relevance of the general awareness regarding use in electronic circuits. In the first paragraph it provides information about the mineral structure and freely admits "(not sure what that means...)"
The same construct might be used in converation when someone is making a fuss about any trend or fad - the intention is to ask
Do you really know what this thing is that everybody's talking about (or are you just talking about it)?
Without using 'anyway' this shade of meaning is not present. For example, just asking:
"What exactly is a quartz crystal?"
or "What is a quartz crystal, exactly?"
Does not carry the implication of a presupposed general level of background interest or awareness in the same way.