As described in SimpleTextFormatting, Ward in the original Wiki needed a subset of HTML to allow simple text to be converted to have HTML elements.
CharacterFormattingRules are formatting clues can exist inside the text anywhere, at the beginning of a line, at the end, or someplace between.
A simple set of text markup is needed to represent various types of character formatting markup that can converted to html.
As described in [WikiBase Text Formatting Rules] there are only 3 CharacterFormattingRules, all for emphasis:
Note that emphasis can be used multiple times within a line, but cannot span across line boundaries.
(Technically links are also a form of character formatting rule, but they are discussed in EasyLinking?.)
The choice of using multiple single quotes was arbitrary, and is not especially obvious or standard outside of Wiki. I've not yet found any precedent for it in other pre-existing text formatting standards (the only thing I was able to find is that '' was typically a close curly quote in typography, paired with `` as a start curly quote). If you weren't told I don't think you'd ever guess it. Even when you know, it's not obvious which is bold and which italic.
This choice was also not oriented for readability of the text, but I don't think that was one of Ward's goals. Reading single-quote emphasis in text isn't easy, especially if double-quotes are are also being used for quoting.
There is evidence that single-quote emphasis was not the best choice as single-quote usage has not survived very well as the Wiki meme has evolved into variants. I don't believe that Ward was aware of the efforts related to doing limited rich text email in SeText -- if he had I think some of his choices may have survived meme variation better.
There are some complexities in the original Wiki emphasis logic, for example, text within doubled single-quotes followed immediately by text within tripled single-quotes is processed incorrectly.
At the time of the release of Wiki in 1995 the Usenet (i.e. News) was quite active and mature. There evolved on there a common convention that asterisks on either side of text signified bold. Italics were not quite as common, nor as standard, but were more typically underline character on either side of italicized words.
Advantages of this solution is that it is short and very readable, and it is obvious what the markup does.
Using asterisks for emphasis has been chosen by the Wiki variants PhpWiki, TwikiClone, SocialText, WikiWriter. Underlines for light emphasis by PhpWiki, TwikiClone, and SocialText.
Unlike the multiple single-quotes used by the original Wiki, using asterisks, slashes, and underlines can be ambiguous. Asterisks and slashes are fairly common in normal text, especially in program text (C uses them for comments, multiplication and dereferencing; filenames and urls use slashes). You can try to disambiguate it (for example by insisting on whitespace to one side), but that gets hairy.
Using asterisks or underlines to emphasize a single word, or a short phrase worked fairly well, however, over longer text had readability problems, and sometimes also parsing problems.
Asterisks were also used by some ParagraphFormattingRules, and sometimes there could be reading or parsing ambigouity between them.