It's not so much a performance thing; people use regular expressions because they are an easy-to-use yet ultra-flexible tool for all sorts of string manipulations. Once you master the syntax, regular expressions allow you to express quite complex string manipulations, and since a regular expression itself is basically a string, you can store regular expressions in variables, files, databases, and a bunch of other things.
Consider this simple regex:
/\bA[a-z]*\b/, which finds all whole words that start with a capital 'A', followed by only lowercase letters (e.g., running it over "And Now All Is Good" would yield ["And", "All"]). Now write a function that does the same (in the language of your choice) using only standard string manipulations. You'll quickly see the advantages of regular expressions.