Minecraft PE won't let me build or break blocks or open chests, how do I fix this?

You're probably on Adventure Mode. Go to the settings, and look for the section titled 'World'. It has a pixelated picture of Earth. The first thing there should say 'Personal Game Mode', access the drop down menu and choose either Creative or Survival mode. Try reloading your world after doing that, and everything should be fine :D

I can explain the differences between Creative and Survival too, if you don't know what they are.

You can try putting yourself on member mode instead of visitor. Visitor mode only permits you to run around, and you can't interact with the world that much. Member mode allows you to break, mine, and everything else that you need to play survival Minecraft. Operator mode gives you commands as well, which is useful if you're building a minigame within Minecraft.