Here's what I've been able to figure out so far:
There are multiple ways to set this up but the easiest one is to move to the lip of the stage and short hop straight up. The timing is as follows:
24-29 Egg Toss
Here is an example of the earliest you can start the egg toss.
It doesn't matter which way you are facing. Other setups include full jump, double jump cancel and even waveland from the platform into double jump cancel but I am not going to figure out the windows for all of those. You also don't have to be extremely close to the lip. As long as Yoshi's nose is above the lip you are fine. These images should help with the range for a neutral jump.
Whichever way you are facing while throwing the egg, you will be facing the ledge afterwards. You will spend one frame intangible and grabbing the ledge, and the following frame tangible and not actionable. Afterwards you can do whatever you want, but you will have some momentum towards the ledge so a neutral jump is not possible without doing something to stop that momentum.
It does not seem possible to do this from the real ledge going onstage. If there is some setup, it is very difficult and not fully intangible.
The reason it works is that the ledge on this particular stage can be grabbed from both sides as long as you have a way to get low enough that your ledge grab box will reach it. In Yoshi's case, his Egg Toss allows him to do that.
In this image, the red and blue boxes (not the large blue box) on either side of Yoshi are his ledge grab boxes, the orange diamonds are his ECBs (and his egg's ECBs are above him) which detect stage collision, and the red line is the ledge.
Overall, not useful but pretty neat.