First of you need PHP. Most linux distribution package PHP in a nice build for running PHP on Apache. But those builds normally does not contain what you need for building extensions. You can eather download PHP youself and compile. Or find out if your dist have a developer package to install that contains phpize.
Next you need ARAPI from BMC Remedy Action Request System. Both include files and library files. It is very important that you have include and library of same version. If you do not know how to extract the ARAPI from the BMC Remedy Request System installation you can try to download here.
Then you need the Remedy Template Library. At the moment RTL seem to be dead. If you download the latest version at SF it only works for ARAPI 7.0/7.1. To get 6.3 support you should use the patch available at RTL on SF.
And finally you need the code from php-arapi. Download it with the link above.
There are two ways to build one is as part of building PHP and the other is to run phpize to be able to make a standalone build. And that is what will be described here.
First you go where the source of php-arapi is and there you run phpize. This will create all the build files needed to build php-arapi.
Next step is to run configure that got created. --enable-arapi will be set by default, but you might use it to set where the ARAPI library. You do that by --enable-arapi=<path>.
Also, before you run configure you should set RTL_DIR to where you have Remedy Template Library.
After running configure there should now be a Makefile. So next is to run make. You will see some warnings during the compile. Some zend functions does not really work well in a C++ compiler. But there should be no problems.
In the modules directory there should now be a file named arapi.so. This file should now be moved into the directory where your other PHP extensions are.
You now only have to make sure all the other .so files needed by arapi.so. On linux it might be enougth to do ldconfig.
Now restart PHP, by restart the web server or whatever is using PHP. You can now use php-arapi.