This answers why you should be doing unit testing.
The 3 videos below cover unit testing in javascript but the general principles apply across most languages.
Unit Testing: Minutes Now Will Save Hours Later - Eric Mann - https://www.youtube.com/watch?v=_UmmaPe8Bzc
JS Unit Testing (very good) - https://www.youtube.com/watch?v=-IYqgx8JxlU
Writing Testable JavaScript - https://www.youtube.com/watch?v=OzjogCFO4Zo
Now I'm just learning about the subject so I may not be 100% correct and there's more to it than what I'm describing here but my basic understanding of unit testing is that you write some test code (which is kept separate from your main code) that calls a function in your main code with input (arguments) that the function requires and the code then checks if it gets back a valid return value. If it does get back a valid value the unit testing framework that you're using to run the tests shows a green light (all good) if the value is invalid you get a red light and you then can fix the problem straight away before you release the new code to production, without testing you may actually not have caught the error.
So you write tests for you current code and create the code so that it passes the test. Months later you or someone else need to modify the function in your main code, because earlier you had already written test code for that function you now run again and the test may fail because the coder introduced a logic error in the function or return something completely different than what that function is supposed to return. Again without the test in place that error might be hard to track down as it can possibly affect other code as well and will go unnoticed.
Also the fact that you have a computer program that runs through your code and tests it instead of you manually doing it in the browser page by page saves time (unit testing for javascript). Let's say that you modify a function that is used by some script on a web page and it works all well and good for its new intended purpose. But, let's also say for arguments sake that there is another function you have somewhere else in your code that depends on that newly modified function for it to operate properly. This dependent function may now stop working because of the changes that you've made to the first function, however without tests in place that are run automatically by your computer you will not notice that there's a problem with that function until it is actually executed and you'll have to manually navigate to a web page that includes the script which executes the dependent function, only then you notice that there's a bug because of the change that you made to the first function.
To reiterate, having tests that are run while developing your application will catch these kinds of problems as you're coding. Not having the tests in place you'd have to manually go through your whole application and even then it can be hard to spot the bug, naively you send it out into production and after a while a kind user sends you a bug report (which won't be as good as your error messages in a testing framework).
It's quite confusing when you first hear of the subject and you think to yourself, am I not already testing my code? And the code that you've written is working like it is supposed to already, "why do I need another framework?"... Yes you are already testing your code but a computer is better at doing it. You just have to write good enough tests for a function/unit of code once and the rest is taken care of for you by the mighty cpu instead of you having to manually check that all of your code is still working when you make a change to your code.
Also, you don't have to unit test your code if you don't want to but it pays off as your project/code base starts to grow larger as the chances of introducing bugs increases.