Go to your igloo, click on your puffle, then a screen will pop up with your puffle on it, you will see 4 options on the bottom, click on the leash on the far right corner. Thats it!
Note: Sometimes this doesnt work, because your puffle needs to have all his energy, rest an health.
If he doesnt, you can play with him, to get his energy high, put him to sleep, or feed him.
Once you do all these, your puffle should be very happy, and you will be able to walk him!