You can unlock new Alleys by collecting trophies: winning matches on any Alley will give you trophies, and losing matches will take trophies away. You can see the total number of trophies you have from all Alleys in the top left corner of the screen.
To unlock new Alleys, you need to increase your total number of trophies. To check how many trophies you need for the next unlock, tap Play to go to the Alley Select screen, then change your current Alley by swiping left or right, or by tapping the arrows to the left and right of your Alley.
You can see how many trophies you will get or lose on the Alley screen, on the Reward tab.
If winning gives you no trophies, but losing makes your trophies go down, then you've reached the trophy limit for your current Alley. This limit can also be seen on the Reward tab of the Alley screen.
To continue collecting trophies, go to another open Alley by swiping left or right on the Alley Select screen, or tap the arrows to the left and right of your Alley, and continue winning!
To summarize: increasing your total number of trophies unlocks new Alleys, and you can add to this total by winning on any Alley that hasn't reached its limit yet.