By placing an object--such as Medli, Makar, an orb, statue, bombs, a pot, etc.--near the perpendicular intersection of a wall and a ledge, it is then possible to clip through a vast majority of the walls in the game.
With Objects other than Bombs
Depending on the object you are using for your clip, there are some slight differences in precision. For example, because pots and bombs are "movable" objects (whilst Link is hanging on the ledge they are being pushed away from him) ledge clipping with such objects can sometimes require frame-perfect, or near frame-perfect, input in order to succeed--in addition, bombs have a time limiting window in which you have to clip. On the other hand, "immovable" objects such, as Medli or Makar, stay stationary during the whole clip. This concept contributes to the reason why clips executed with the help of Medli or Makar don't require pause buffering. Also, if the trick is failed, the clip will still work fine without having to replace the character in position.
There are a few unique ledges in the game that behave differently from the type described above. One of the prime examples are the door frames in the Tower of the Gods. These ledges simply depend on the proper angle and do not require any additional object in order to clip. For more information, click here.
For some ledge clips (particularly the one to skip the DRC boss key), pause buffering is a more accurate, as well as a safer way to insure that you will clip into the desired wall. When Link grabs a ledge, he bends his legs and as you advance frames they will straiten out. There is a frame where Link's legs are still slightly bent, and the next they are completely strait. This is what you are looking for when pause buffering. Depending on the ledge, either both frames, or only the latter will work (with a 2-3 frame window). On this frame press away from the wall to clip as you would normally.
Both pause-buffered & non-pause-buffered methods are demonstrated in the video below.