You'll be lucky if this doesn't get downvoted to hell, but I'll try and give you at least a basic run-through.
There are two ways to approach this. AJAX or PHP. The PHP method is a little more complicated to wrap your head around (at least for me anyway).
PHP
First, you should sit down and REALLY read the Wikipedia API manual and sources. The people behind Wikipedia have put a ton of effort into it and wouldn't have done so without giving you information on how to use their system. Don't be intimidated--it's really not that hard.
Second, after you've read the API, you'll know what this url means.
http://en.wikipedia.org/w/api.php?format=json&action=query&titles=Main%20Page&prop=revisions&rvprop=content
A method that might work without using CURL, which can be very confusing, is file_get_contents().
So set the query string parameters for the api, and use them like so:
$api_call = file_get_contents('http://en.wikipedia.org/w/api.php?format=json&action=query&titles=Main%20Page&prop=revisions&rvprop=content');
$api_data = json_decode( $homepage );
Now you should have an array that you should easily be able to manipulate and place into your site.
jQuery/AJAX
Way easier in my opinion, and you have the added bonus of some user manipulation:
$.ajax({
type: 'GET',
url: 'http://en.wikipedia.org/w/api.php?format=json&action=query&titles=Main%20Page&prop=revisions&rvprop=content'
}).done(function(data){
var d = $.parseJSON( data );
$('div.data-holder').html( 'foo'+d.bar );
});
None of this is tested. Just meant for a general idea. Hope this helps.