I'm currently working on a cutting board with a rectangular cut out for a handle. I used a jig saw to cutout the handle, and need to sand the edges of the cut.
Here's what I would suggest.
If you're still not done with the design, I would consider making the ends of the cutout rounded, similar to the image below. If this is not to your taste, please move on past the line break below.
This serves two purposes: first, the handle is more comfortable to hold, and second, it's easier to smooth with no sharp corners.
If you have a router, I would use a roundover bit on the handle to smooth the transition. If you don't have a router, continue on to the next step.
Now, if you want to keep the sharp inside corners like you have now, I would recommend using thin sanding belts to reach inside the handle opening. You would work the sanding belt in a back-and-forth motion, like you were flossing your teeth.
They also make thin belt sanders that can do this the powered way. Harbor Freight is not my favorite tool company, but their tool is shown in the example below.
Please also read this related Question where the OP has an issue similar to yours. @Graphus's answer involving using a scraping chisel would work here.