How could a system be zero-knowledge?

In the simplest models for symmetric (reversible) encryption, you just need a decryption key - which can be as simple as a common password.

For the system to be zero-knowledge, the encryption must be done on the client's system and ONLY encrypted information passed to storage provider. This way the storage provider technically has the data stored, yet it has no way to access the clear text (human-readable un-encrypted version) because it doesn't have the key.

Note that in the old-days, we did this with simple FTP and "password protected ZIP files".

The weakness of this system is then that if the decryption key is just a common password then the storage provider (or anyone that intercepts the data) could trivially brute-force crack it and get the clear-text.

So, to make this system even more safe and robust, usually a public-private key system is used, again with the client's private key always being left on the client and NEVER given to the storage provider. Any storage provider that accepts or can access the decryption key is not zero-knowledge, by definition.

As you point out, this could be inconvenient - if I store information with the provider and want to access it at home, I have to manage the key myself. Many tools exist for this, from "keyrings" to secure USB dongle's to biometric storage devices with TPM chips, etc. But yes, you have to carry the key around with you or store it on some other provider's server, with all the potential dangers and inconvenience that has.

The simplest way to understand the systems is just to compare them to a keyed safe. You put in things you don't want anyone to touch, and you lock the safe. You give the safe to a storage company, and then you have the security of knowing the storage company only has access to your safe - not it's contents. If you also have the storage company store the keys to the safe, well then obviously they could trivially access it anytime they want.

The price of security and privacy is very often inconvenience - sometimes tremendous inconvenience. You just have to decide if what you gain is worth the price you must pay in time and potential insecurity (if you lose your key that data is pretty much gone forever, no one can get it back for you).