CSS is not a script, so you dont have to “execute” it in the sense of script execution.
Basically a <link> tag created on the fly and appended to the header should suffice, like
$('<link/>', {
rel: 'stylesheet',
type: 'text/css',
href: "https://stackoverflow.com/questions/14919894/path_to_the.css"
}).appendTo('head');
or
var linkElem = document.createElement('link');
document.getElementsByTagName('head')[0].appendChild(linkElem);
linkElem.rel="stylesheet";
linkElem.type="text/css";
linkElem.href="https://stackoverflow.com/questions/14919894/path_to_the.css";
if you want to do it without jQuery.
The browser will respond to the change in the DOM and update your page layout accordingly.
EDIT:
I have read that old Internet Explorer has trouble with this, and you might need to do it like in answer to make it work:
https://stackoverflow.com/a/2685639/618206
EDIT2:
Reading the file content and putting it inline (between <style> tags) is also a valid solution, but that way the CSS will not be cached by the browser.