Imagine that the user has a password "stack overflow" entered during registration on the website. The website will split the password in two, the first part being always four characters long:
Part 1: "stac"
Part 2: "k overflow"
Then some random characters are appended to the first part, and prepended to the second part, given that the prepended part is of a given length, let's say 6:
Part 1: "stacd9u8BHaI1"
Part 2: "hBy47k overflow"
The first part is salted and hashed on a first server and stored in the database. The second part uses a different salt, different hash, and a database on a different server.
Imagine a hacker gains access to the first database and successfully finds, for example through brute force, the password "stacd9u8BHaI1". The hacker tests this password on the website, and notices that it doesn't work. He may be discouraged by that, and abandon the tentative of finding other passwords. It's supplementary security, given that it's not extremely strong, but still valuable if it doesn't cost too much in terms of resources compared to the ordinary, single server authentication.
Isn't it easy to discover what part of password is stored?
Actually, no. It's possible, but not easy. If you find, through brute force, that the original password is "hBy47k overflow", you may guess that the beginning was randomized, but it's not straightforward. Random characters may also be adjusted accordingly: for example for passwords which contain only words, a word may be prepended/appended, instead of random characters:
Part 1: "stac battery" // Appending a random number of characters
Part 2: "change k overflow" // Prepending 7 characters
Part 1: "stac white unicorn"
Part 2: "global k overflow" // Still 7 characters
You actually need to brute force several passwords to guess how much prepended/appended characters are there.
When it fails?
There is a case when this technique fails miserably: if you have an account on a hacked system, you already know your password, so by bruteforcing the two parts, you can easily see what parts were preserved. Making the number of characters in the first part and the number of characters prepended to the second part dependent on an element relative to the password itself makes it more difficult to understand the system, but still quite doable.
Do I need to implement this system right now?
No. Security which relies on the fact that you keep secret the way you store passwords is weak. Anybody who has an access to the source code knows all the elements.
The split the password technique has a tiny advantage, but still doesn't worth to be implemented in every website. Just use PBKDF2 with correct parameters, and you'll be fine. Even if you're a bank. I'm pretty sure my bank stores my password in plain in the database, as well as most websites you use daily. That sucks.