Sticker Packs contain Party Stickers which are used to level up parties. You can win them from various sources such as the Spin and parties themselves or as a bonus reward from other features.
There are 2 types of Sticker Packs; green and purple ones. Both of them can be bought with cash on the party popup or can be won from the spin.
The green packs are normal ones which provide random stickers. This pack may cointain duplicates, but you'll earn a reward if it's a duplicate sticker; so there's no losing!
The purple packs will give you unique stickers. This means you will definitely get a new one that you don't already have!
As you collect stickers, you'll see that they are accumulated in your parties popup, and can be used from there to level up parties.
When you have all four parts of the same color of stickers for a party, you'll level it up!