We write tests to verify the correctness of a program's behaviour.
Verifying the correctness of a program's behaviour by inspecting the content of output statements using your eyes is a manual, or more specifically, a visual process.
You could argue that
visual inspection works, I check that the code does what it's meant to
do, for these scenarios and once I can see it's correct we're good to
go.
Now first up, it's great to that you are interested in whether or not the code works correctly. That's a good thing. You're ahead of the curve! Sadly, there are problems with this as an approach.
The first problem with visual inspection is that you're a bad welding accident away from never being able to check your code's correctness again.
The second problem is that the pair of eyes used is tightly coupled with the brain of the owner of the eyes. If the author of the code also owns the eyes used in the visual inspection process, the process of verifying correctness has a dependency on the knowledge about the program internalised in the visual inspector's brain.
It is difficult for a new pair of eyes to come in and verify the correctness of the code simply because they are not partnered up with brain of the original coder. The owner of the second pair of eyes will have to converse with original author of the code in order to fully understand the code in question. Conversation as a means of sharing knowledge is notoriously unreliable. A point which is moot if the Original Coder is unavailable to the new pair eyes. In that instance the new pair of eyes has to read the original code.
Reading other people's code that is not covered by unit tests is more difficult than reading code that has associated unit tests. At best reading other peoples code is tricky work, at its worst this is the most turgid task in software engineering. There's a reason that employers, when advertising job vacancies, stress that a project is a greenfield (or brand new) one. Writing code from scratch is easier than modifying existing code and thereby makes the advertised job appear more attractive to potential employees.
With unit testing we divide code up into its component parts. For each component we then set out our stall stating how the program should behave. Each unit test tells a story of how that part of the program should act in a specific scenario. Each unit test is like a clause in a contract that describes what should happen from the client code's point of view.
This then means that a new pair of eyes has two strands of live and accurate documentation on the code in question.
First they have the code itself, the implementation, how the code was done; second they have all of the knowledge that the original coder described in a set of formal statements that tell the story of how this code is supposed to behave.
Unit tests capture and formally describe the knowledge that the original author possessed when they implemented the class. They provide a description of how that class behaves when used by a client.
You are correct to question the usefulness of doing this because it is possible to write unit tests that are useless, do not cover all of the code in question, become stale or out of date and so on. How do we ensure that unit tests not only mimics but improves upon the process of a knowledgeable, conscientious author visually inspecting their code's output statements at runtime? Write the unit test first then write the code to make that test pass. When you are finished, let the computers run the tests, they're fast they are great at doing repetitive tasks they are ideally suited to the job.
Ensure test quality by reviewing them each time you touch off the code they test and run the tests for each build. If a test fails, fix it immediately.
We automate the process of running tests so that they are run each time we do a build of the project. We also automate the generation of code coverage reports that details what percentage of code that is covered and exercised by tests. We strive for high percentages. Some companies will prevent code changes from being checked in to source code control if they do not have sufficient unit tests written to describe any changes in behaviour to the code. Typically a second pair of eyes will review code changes in conjunction with the author of the changes. The reviewer will go through the changes ensure that the changes understandable and sufficiently covered by tests. So the review process is manual, but when the tests (unit and integration tests and possibly user acceptance tests) pass this manual review process the become part of the automatic build process. These are run each time a change is checked in. A continuous-integration server carries out this task as part of the build process.
Tests that are automatically run, maintain the integrity of the code's behaviour and help to prevent future changes to the code base from breaking the code.
Finally, providing tests allows you to aggressively re-factor code because you can make big code improvements safe in the knowledge that your changes do not break existing tests.
There is a caveat to Test Driven Development and that is that you have to write code with an eye to making it testable. This involves coding to interfaces and using techniques such as Dependency Injection to instantiate collaborating objects. Check out the work of Kent Beck who describes TDD very well. Look up coding to interfaces and study design-patterns