This St. Patrick's Day scrapbook paper design is called Fluttery Shamrocks for good reason, because it features shamrocks falling on a soft background of aqua and subtle scrollwork. Click the link in this sentence for many scrapbooking papers that coordinate with this design, as well as other coordinating sets of free St. Patrick's Day scrapbook papers.
I don't know how the shamrock became a symbol of St. Patrick's Day. Perhaps it was because Ireland -- the home of St. Patrick -- is called the Emerald Isle and is covered with shamrocks. That's a reasonable assumption! Whatever the reason, shamrocks are instantly recognizable as symbols of St. Patrick's Day. So it was a natural thing to add them to this scrapbook paper design and many of the others you can find on the link in the previous paragraph.
You might also enjoy this free St. Patrick's Day scrapbook journaling card. And make sure to see the sidebar for other free scrapbooking supplies!