Sounds like you are trying to 'sell' unit testing in your company. (Good for you!)
I am going to risk being hated here, but it sounds like you youself might not
be 100% clear on all the benefits of writing unit tests. As other
answers have pointed out, there are many reasons unit tests are extremely
useful (sometimes even necessary) other than catching bugs that the compiler
did not catch.
I suggest getting familiar with all the benefits and designing my demo accordingly.
Here are some quick things I would talk about/demo (not in any special order):
Quick validation that unit of code works without having to actually run it
in its intended environment. This is extremely crucial when it takes 5 min
to build your project, 10 min to deploy it and half an hour to try to
find the scenario when the code actually runs to validate that it 'works'.
Validation of code that would be extremely hard to test in its intended
environment. A huge percentage of the code we write is handling error conditions
or conditions that only happen in certain strange circumstances. These conditions
are very hard to reproduce in real life, therefore require special attention
to test.
Validating 'library' code. If you are writing code that others consume, your only
way of validating would be to write another 'project' that consumes and drives
it. Unit tests give you a high confidence that your code works without having
to use a project that drives your code to test it or to write a 'tester' app
yourself.
TDD. Test Driven Development is a fairly new way of thinking about
unit tests. When exercising TDD developers write the tests first,
then code afterwards. TDD can improve your code by only writing code that satisfies passing your tests. There is much debate about this topic
and myself used it extremely successfully in certain situations, but not so
successfully in others (context dependent IMHO).
Tests document your code. Many times tests are the most updated documentation
on how your code is supposed to behave. How many times do you look for an already
existing piece of code that exercises a class/method just to see how you are
supposed to use it? Well unit tests give you just that.
Code breakage. Yes, unit tests do catch bugs. However I find that they most often
catch bugs during development. I write the code, then I turn around to write the
test and this is the phase when I catch most bugs.
I am sure there are more to mention here so you should do more research
before your training. One more thing. It is hard to find examples when unit tests
'saved the day in a big way' because they catch bugs during development and
very silently most of the times. You only really understand their benefits
when you have been writing them for a while to see how they save the day
(on a daily basis).
I would compare them to exercising. It is hard to find a case when daily
exercise saved someone from having cancer or a heart attack.
But you do know deep down that it is necessary to live a long and healthy life.