"Game progress" (and with it story progress) usually works through a conjugation of abilities and flags. Depending on which abilities you get and which flags you manage to set, the game will seem to be at a certain point in the story.
Abilities are exactly what they sound like. In Zelda, they're mostly items you pick up that let you do things like set objects on fire, shoot arrows, or become a wolf. These abilities let you access things you perhaps shouldn't be able to access yet, meaning you'll be experiencing the story out of order and getting a fragmented experience.
Flags are simply values that dictate what has happened in the game. For example, when you pick up the master sword, the "has gotten master sword" flag is set to true. When the game loads different areas, it checks for certain flags - for example, if you've talked to a certain person about something, a flag may be set and a guard moves from an opening. Or perhaps if you've killed a certain boss, the townspeople will be happy about it. Doing things that set flags too early might seriously mess up the game flow and make you miss things completely, in addition to things like leaving half the game world in one story state, while the other half is in another.
The kinds of cheats that won't harm the story progression is things like "your sword does triple damage" or "run faster" (assuming you don't run somewhere you shouldn't be), as those just amplify the things you can already do. Depending on how the cheat works, things like "have this item" (without explicitly getting it) may also not mess with the flags, but they'll often let you have abilities you shouldn't which gets you into weird situations. When you do go to pick up that item nothing terrible should happen, but you will still have a very strange experience and it probably won't be much faster.