If you’d like to move your iExplorer backups that you see in the Browse iTunes Backups section, for both iExplorer 3 and iExplorer 4, you could move the backup folders for each backup to the new computer using any external storage medium. The Backup section is just displaying the data within iTunes backups on your computer which are stored in iTunes’ default backup folder location in the MobileSync/Backup (see the full folder path in the article at the link below). We do not collect or store any of your personal data online for your privacy, and iExplorer, the app itself, doesn’t store any data within it or in its program folder. It acts more as a window showing the data on your device or what’s in an iTunes backup on your computer.
Then, if you don’t want iTunes on the new laptop, you could put each of the backup folders in any folder on your computer and point iExplorer to read those backups without even needing iTunes installed on the laptop. Here’s an article that explains how to point iExplorer to read iTunes backups not stored in the default location that iTunes uses:
Can I choose which folder iExplorer uses for creating or reading Backups?