In order to expand your town, you will need to purchase plots of land.
This can be done by tapping on the locked plot of land. 

Take note that you may need to gain some additional levels to unlock areas further from the starting point. These areas will have an icon displaying a lock. Whilst areas that can be unlocked, will show a different icon above the locked plot of land. 

When tapping on the plot of land you wish to unlock, you will see that you need to provide specific items as well as currency. These required items can vary. 

In the image above, these items can be obtained from the Treasure Dealer which is accessed by tapping on the bell icon as shown below. 

More infos on using the Treasure Dealer can be found here: