How can I repair the drywall by this switch?

Chip out CAREFULLY all the way to the molding and then build your own wood frame around the switch box...You gonna want to replace a rectangle piece of the drywall anyway...so a little more destruction before the construction will make it look better and last more.

Drywall repair clips

How big is the hole? I cannot see the the entire opening in the photo. There are wire mesh patches that will cover large holes to repair. Most home improvement stores carry the wire mesh ( do not use the fiberglass mess because you need the stiffness of the wire). I believe they make them as large as 8"x8", They have an adhesive backing applied to the back to help it stick, it will take several applications of spackle to keep it from cracking while drying. Best of luck

I would go with Athenaindigo's suggestion, it makes sense to secure the switch box to something and not just have it hanging like that, and will also make the repair easier so it can be done right.

Here is a perfect video for your repair. Working as a maintenance supervisor for a nursing home I and my crew have used the California patch all the time. For large holes we used the wood strip backing process. Walls were drywall over metal studs which was code for nursing homes.

https://www.lowes.com/projects/repair-and-maintain/patch-and-repair-drywall/project

looks like you could take out the whole box and work from the inside. Attach a piece of dry wall with Gorilla glue and put box back and than refinish

I would turn the breaker to the switch off remove it and replace the wallboard and then reattach the switch box. There is no quick fix here with a patch.

it looks like the whole box is attached in behind drywall ,the back part looks screwed to something . Does the electric box seem firm to touch, if so then just cut out a opening for it. If not then go look at electric box for holes to screw to back wall ?? my guess from photo ?!