I want to understand the main reason for ending the support. Are there any security issues in the older versions because of which they don't want to continue or maybe a performance issue or bugs which they don't want to fix?
It's mainly because maintaining many different versions is a lot of work, for little benefit.
I'm not aware of any significant performance or security issues with older versions of Laravel. Of course, this may not be true forever, but it's probably going to be true for a long time.
What could be the disadvantages of keeping an older version?
Older versions are still available. It's just that they don't receive the same level of support.
The reason why I'm asking this is that my organization has most of the projects in the older versions and I'm not sure if I should upgrade them to the newer versions. I need a solid reason to upgrade them other than the obvious structural and architectural improvements.
If a project is still being actively developed, then it may be worth upgrading (I probably would, but I'm sure it depends on the project).
If your project is no longer being developed and is feature-locked, then it's probably not worth upgrading.