“Drilling out screws” is a broad term. It happens when a screw cannot be removed in the normal manner. Usually by this time the drive feature: slot, socket, hex head, is damaged beyond use, or even the entire head might be broken off from the shaft. In this case, the male and female heads are locked together due to rust, corrosion, the improper use of an overly strong bonding agent. In some cases, a screw might have bottomed out in a too shallow hole leaving its bottom swedged in place with overly aggressive tightening. A frozen screw can happen when a PERMAMENT thread-locking compound is incorrectly applied where a reversable (REMOVABLE) compound should have been used.
For sufficiently large screws, it might be possible to drill a specified diameter hole into the screw as close as possible to the center so that a tapered helical screw extractor can be driven into the hole in the loosening direction. Sometimes this works. Other times it doesn’t.
https://en.wikipedia.org/wiki/Screw_extractor
There are several alternatives, some of which can only be done in well-equipped shop. On a large enough workpiece, such as a stamping die, the offending screw can be drilled out using a larger drill diameter for the tap of a larger screw. Then the hole is plugged with a the larger sized glued-in threaded rod or screw of a material soft enough that it can be drilled and tapped with the required thread. The plug must then be leveled flush with the surrounding area. Then the hole of the correct size is carefully positioned, drilled and tapped to restore it.
For particularly tough alloy screws, a carbide drill might be required. If so, this should be done on a drill press or milling machine with fine feed so that the drill can make its way slowly through the material without destroying itself. Larger sized holes might require step drilling. That’s where the center of the hole is made with a smaller drill, and then the hole is drilled to the required size with a larger drill. Taking out the center first with a smaller drill greatly reduces the resistance when drilling the final diameter.
Here’s an example of drilling, plugging, and retaping a damaged hole. I recently received a manufacturer’s sealed box containing a Beseler 8086 enlarger lens board "Mounted 39 mm Leica Flange" via eBay purchase. Both the lens board and flange are aluminum alloy.
The flange has three equally spaced screw holes countersunk for flathead screws to mount the flange to the lens board. The screws installed were #2-56 round head screws, OD = 0.086”/2.18 mm. The heads stuck upward so that a lens could not be mounted. One screw head was particularly high. When I disassembled the flange from the board, the underlying problem was evident. Two of the screw holes were positioned correctly. But the third hole was nearly half a screw diameter out of position. That forced all three round head screws badly out of position in assembly. Round head screws prevented mounting a lens. The botched assembly was unusable.
My solution was to drill out the useless out-of-position hole and tap it for a #4-40 screw, diameter 0.110”/2.79 mm. It wasn't possible to use a larger screw because the required tap drill would have broken out into the large clearance hole for the locating ridge of the flange (holes close to the edge).
I used a brass screw because it’s soft enough to drill and tap. I glued the brass screw into the lens board with Loctite 620 PERMANENT bonding compound and left it overnight for the bond to reach maximum strength.
Then I used a fine-tooth hacksaw blade to trim off the excess screw length. I finished with a Dremel using abrasive wheels to make the brass plug flush to the surrounding surface. I installed the flange with two proper flat head screws and transferred the missing hole position from the flange into the brass plug. Due to the overlap, the resulting #2-56 tapped hole was partly into the aluminum lens broad and partly into the brass plug and well-centered in the screw hole of the mounted flange.
It’s now perfectly usable and the flange covers the repair. Such a repair is best done on a drill press or milling machine with proper work-holding device. With a milling machine, a small-diameter end mill could have been used to mill the plug flush to the surrounding area. Much easier.