You make progress toward your next level each time you place a bet and Spin in any of the slot machines. The higher the bet, the more progress is made. After you have made enough progress, your level increases.
New Slot machines are unlocked every few levels. You should be able to see the requirements for each machine in your lobby (slot machine selection page). Once you reach the level specified, the new game will automatically be unlocked for you. If you play on the mobile app, you may be required to download the new machine before you begin playing.