TLDR: Using password managers promotes best practices that increase your password security by allowing you to choose impossible to crack passwords and secure them by only one password. Good analogy is having all your eggs in one basket and putting that basket in a secured locker in a Bank.
This is a bit divergent but necessary to understand the argument.
You need to understand how passwords are stored by the website. In modern applications no one stores the passwords in plain-text (which means storing the password as-is). The websites will store a hash (hash is a value derived from your password). Hashing a password is one-way-street. You can not find out the password from the hash.
Now you may ask how the passwords get leaked? Now there are some fellows out there who have precomputed hashes of all the common passwords. Such a dictionary of passwords and their hashes is called rainbow table. Now if you have access to this list then you can lookup the hash in the list to get the password. Now there are ways to delay this but someone adamant can still find your password.
Now the passwords which are easy to remember like cool-hack-123 are easy to remember but are also easy to crack. And the passwords which are difficult to hack are impossible to remember on scale (by scale I mean there are hundreds of passwords that need to be remembered). So people resort to using the same difficult password everywhere and trust the websites to keep their servers safe. Now doing this means that if one of the websites leaks the data and someone finds your password then your accounts where you used same password are at risk.
The only way to mitigate this is to use different difficult passwords. Which is possible but impractical because the day you forget one password you will set it to something you can remember. This is where the password managers come in. These applications use state of the art security algorithms and processes to store your password information. They take over the job of remembering passwords. Once you start using them you are not bound by any limitations on password selection. You can select an Uber random 24 character password and rest assured that it will be there when you need it.
There are algorithms like AES which when used with 256 bit keys are proven to get eons to crack. There is a public key infrastructure which can be employed to verify authenticity of the data and securely communicate it without sharing the passwords used for encryption. Now these services have best cryptographic minds working for them who have carefully selected the tools (from an arsenal of time tested algorithms) to secure customer information.
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