The {{" ' "}} (or{{Double+single+double}} in long form) template is for use a) at the beginning of a "double-quoted" quotation that begins a 'single-quoted' quotation that in turn begins an enclosing "double-quoted" quotation, or b) at the end of a "double-quoted" quotation that ends a 'single-quoted' quotation that in turn ends an enclosing "double-quoted" quotation. Its purpose is to slightly space the single and double quotation marks apart for legibility, without actually inserting a space character. It does this with CSS, and does so because the insertion of extraneous space characters of any kind would be a violation of the semantic integrity of the web content of the article or other page in which it appeared.
Example (with highlighting added for clarity)
Using the template:
Kwan was quoted as saying, "She disputed his statement that 'Voltaire never said "I disapprove of what you say, but I will defend to the death your right to say it.{{" ' "}}
Kwan was quoted as saying, "She disputed his statement that 'Voltaire never said "I disapprove of what you say, but I will defend to the death your right to say it."'"
For comparison, here is the completely unspaced version:
Kwan was quoted as saying, "She disputed his statement that 'Voltaire never said "I disapprove of what you say, but I will defend to the death your right to say it."'"
Kwan was quoted as saying, "She disputed his statement that 'Voltaire never said "I disapprove of what you say, but I will defend to the death your right to say it."'"
And here is the version with a semantically incorrect non-breaking space character:
Kwan was quoted as saying, "She disputed his statement that 'Voltaire never said "I disapprove of what you say, but I will defend to the death your right to say it." ' "
Kwan was quoted as saying, "She disputed his statement that 'Voltaire never said "I disapprove of what you say, but I will defend to the death your right to say it." ' "
When rendered, the non-breaking-spaced version looks similar to or the same as the template version, but improperly mixes content and presentation.
Mozilla/Firefox note
On some Mozilla-based browsers, the spacing is not perfectly even on both sides of the single-quote. There isn't anything we can do about this without making it look equally imperfect in most or all other browsers. This is a Mozilla bug, not a MediaWiki bug.
