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I will make a guess that your server is running CloudLinux. Due to a bug in CloudLinux the filesystem cache (which is shared with the total memory available to your account) is not cleared fast enough. As a result, any operation involving a large number of files – such as backing up or restoring a site – ends up eating up all your available memory.

There are two workarounds.

The first workaround is slowing down the extraction process. If you are doing a restoration from inside Akeeba Backup this does not apply to you. It only applies when using Kickstart. When you run Kickstart set the Minimum Execution Time to 5 and the Maximum Execution Time to 1 (yes, max is less than min, this is not a typo!). This makes the extraction 80% slower which gives enough time to CloudLinux to clear the filesystem cache, freeing up memory (resources).

Note: We have plans to add these advanced settings to the integrated restoration as well in the next version of Akeeba Backup. When we do, the workaround above will also apply to restoring backup archives from inside Akeeba Backup itself.

The second workaround is a bit more "brutal". You need to extract the backup archive on your computer using Kickstart, using a local server such as XAMPP, MAMP or WampServer. When the extraction is complete do not click on "Run the restoration script". Instead, upload all of the extracted files (except kickstart.php and the backup archive itself) to your server. Moreover, delete the .htaccess file from your site's web root. Then access the /installation/index.php URL on your live site to complete the restoration. This works because the bulk of file operations is done by FTP which does not trigger CloudLinux' filesystem cache handling bug.

The screenshot is super low resolution and I can't read the text. But I know what the message reads since I was the one who wrote it ;) Please read it carefully. It is not an error message. It tells you what to do if you get an error message. So, your answer is on your screen.