A lot of times some quick task comes up that isn't part of the main software you are developing. Sometimes the task is one off ie compare this file to the database and let me know the differences. It is a lot easier to do text parsing in Perl/Ruby/Python than it is in Java or C# (partially because it is a lot easier to use regular expressions). It will probably take a lot less time to parse the text file using Perl/Ruby/Python (or maybe even vbscript cringe and then load it into the database than it would to create a Java/C# program to do it or to do it by hand.
Also, due to the ease at which most of the dynamic languages parse text, they are great for code generation. Sure your final project must be in C#/Java/Transact SQL but instead of cutting and pasting 100 times, finding errors, and cutting and pasting another 100 times it is often (but not always) easier just to use a code generator.
A recent example at work is we needed to get data from one accounting system into our accounting system. The system has an import format, but the old system had a completely different format (fixed width although some things had to be matched). The task is not to create a program to migrate the data over and over again. It is to shove the data into our system and then maintain it there going forward. So even though we are a C# and SQL Server shop, I used Python to convert the data into the format that could be imported by our application. Ultimately it doesn't matter that I used python, it matters that the data is in the system. My boss was pretty impressed.
Where I often see the dynamic languages used for is testing. It is much easier to create a Python/Perl/Ruby program to link to a web service and throw some data against it than it is to create the equivalent Java program. You can also use python to hit against command line programs, generate a ton of garbage (but still valid) test data, etc.. quite easily.
The other thing that dynamic languages are big on is code generation. Creating the C#/C++/Java code. Some examples follow:
The first code generation task I often see is people using dynamic languages to maintain constants in the system. Instead of hand coding a bunch of enums, a dynamic language can be used to fairly easily parse a text file and create the Java/C# code with the enums.
SQL is a whole other ball game but often you get better performance by cut and pasting 100 times instead of trying to do a function (due to caching of execution plans or putting complicated logic in a function causing you to go row by row instead of in a set). In fact it is quite useful to use the table definition to create certain stored procedures automatically.
It is always better to get buy in for a code generator. But even if you don't, is it more fun to spend time cutting/pasting or is it more fun to create a Perl/Python/Ruby script once and then have that generate the code? If it takes you hours to hand code something but less time to create a code generator, then even if you use it once you have saved time and hence money. If it takes you longer to create a code generator than it takes to hand code once but you know you will have to update the code more than once, it may still make sense. If it takes you 2 hours to hand code, 4 hours to do the generator but you know you'll have to hand code equivalent work another 5 or 6 times than it is obviously better to create the generator.
Also some things are easier with dynamic languages than Java/C#/C/C++. In particular regular expressions come to mind. If you start using regular expressions in Perl and realize their value, you may suddenly start making use of the Java regular expression library if you haven't before. If you have then there may be something else.
I will leave you with one last example of a task that would have been great for a dynamic language. My work mate had to take a directory full of files and burn them to various cd's for various customers. There were a few customers but a lot of files and you had to look in them to see what they were. He did this task by hand....A Java/C# program would have saved time, but for one time and with all the development overhead it isn't worth it. However slapping something together in Perl/Python/Ruby probably would have been worth it. He spent several hours doing it. It would have taken less than one to create the Python script to inspect each file, match which customer it goes to, and then move the file to the appropriate place.....Again, not part of the standard job. But the task came up as a one off. Is it better to do it yourself, spend the larger amount of time to make Java/C# do the task, or spend a much smaller amount of time doing it in Python/Perl/Ruby. If you are using C or C++ the point is even more dramatic due to the extra concerns of programming in C or C++ (pointers, no array bounds checking, etc.).