How do Wikipedia and other sites automatically tranform what a user types inbetween certain tags into certain types of links?

I was just reading this Wikipedia page out of curiosity, and I was wondering if a programmer or web dev could explain to me how a certain function of it and other websites (including Stackexchange) work on the backend.

If you want to link to an article, but display some other text for the link, you can do so by adding the pipe "|" divider (SHIFT + BACKSLASH on English-layout and other keyboards) followed by the alternative name. Place the article you want to link to before the pipe divider, and the text to be seen after the pipe divider. Two examples:

1. :[[Fox Broadcasting Company|Fox]] In the finished article, the text will read 'Fox', and link to the article called 'Fox Broadcasting Company'.

How do wikipedia and Stackexchange know to turn something in between brackets ""[[]]" into a link to a wikipedia article? Is it a server-side controller like I learned about in my .NET beginner's course?