I've seen two uses.
1. Security as already mentioned. Some RDBMS' allow you to secure individual columns but others do not. If your RDBMS does not implement colum level security, you can use 1-1 relationships to add higher levels of security.
2. When you have an entity that can take multiple forms and you need to keep specialized data for each entity type. You might find this in an application that includes Employees, Customers, and Vendors. In some places in the application, the entities are interchangeable and so you want to be able to have one table to join to but since each type has special attributes, you create an entity table with a type field and all the common attributes and three one-side tables with the specific attributes for each type. For relationships to contacts and addresses, ALL relationships go to tblEntity rather than to the specific type tables.
The 1-1 is extremely rare and frequently mis-used by people who have their excel hat on and think that repeating groups like year, month, day, etc should be columns rather than rows and so they often run out of room with Access' limitation of 255 columns. Or sometimes they just think they are being organized by grouping attributes into multiple tables. There is a horrendous example of this mistake in production in my state. A vendor who shall remain nameless created a survey application and made each question a separate column in a table. So rather than a standard survey where questions are rows and you have junction tables to put them into groups for reporting and display, they separated the questions into 100 tables. Talk about job security. Want to add a question, pay for them to modify the table and any associated forms and queries. They actually hard-coded the questions onto web pages!!!! They also charged Three million + and took almost three times as long as they estimated to build the app which turned out to be over two years. Plus because they saw this as a real time requirement, the user companies had to purchase iPads and hotspots for every care manager who worked in the field. My bid to do the project using Access with offline collection using their existing laptops without requiring them to bring their own hotspots was so low by comparison that the PTB thought it was a joke. I guess the joke was on me since instead of getting paid millions of dollars for a project that should have cost under $100,000, I got to pay for the boondoggle with my tax money.
I still don't know whether the application designers were just incompetent or simply decided to make the most inefficient design possible in order to extract the largest possible fee along with a long term maintenance contract.