I have to say I have never had an problems with this, but I know it's confusing.
A CD is a storage media that holds files which cannot be altered in anyway, and cannot (generally) be deleted. They are not like hard disks where we can add, remove, rename, and so on, as and when we wish. In fact the way I see it, calling these files "Read Only" is a little misleading, not by you but by the industry and by us.
Think of music CDs that we buy from shops. We cannot change that CD in any way. The normal CDs we use for copying data to are similar in a way. Once they are copied, (burned), to a disk, they are permanent. I won't go into CD (RW), (CD Read/Write that can, in fact, manipulate files), because they are slightly different.
So, the files you copied on to CD are there permanently and cannot be changed. That is why they are called "Read Only".
But when you get your new computer, you should be able to simply copy the files from the CD onto the new computer's hard drive. They should then lose their "Read only" status, but if not you can soon change them individually.
HOWEVER I do see one possible complication. Don't throw that old computer away immediately you get the new one because you may need it. When we record, (or copy), data on to a CD, we have the choice of closing or finalizing the session on the CD, or leaving the session open. They are called "Closed CDs" or "Open CDs".
A closed CD can be read by any computer. But a closed CD cannot have any more data copied onto it.
An open CD means more data can be added to it, but no other computer will see that data. Only the computer that was used to copy the data onto the CD can see those files.
You may need to insert the CD, (or CDs), back into the original computer and use the software you used to copy the data to them to close the session. Then it should be able to be used by any other computer.
That is why we now tend to not bother with transferring files onto CD but instead use USB memory sticks or external USB drives. Apart from having much more capacity, they do not suffer from this closed/open problem. And also, coping data is so much easier. Simply Copy/Paste onto the memory stick or external drive.
I hope that helps.