iExplorer, the app itself, does not contain any of your data inside the app or online. It’s more of a window showing the data on your device or in an iTunes backup. So, if you haven’t already, just be sure to go through each section of iExplorer and use the Export buttons to make copies of that data on your computer and outside an iTunes backup, and then you could move those files to the new computer via an external hard drive or any other way you’d like to transfer the files. They are just like any other computer files at that point which you can do whatever you’d like with.
If most of this data is still on your iPhones, iPads, and iPods, of course you could also just install iExplorer on the new computer and use it with your devices to transfer the data from the device to your new computer.
If you are wanting to transfer entire iTunes backups, you could just copy the contents of the MobileSync folder that iTunes stores them from the old computer to the new one and just be sure to copy them to the exact same location. You can find out where they are stored in the AppleCare article below: