The skip begins one screen up and right from Richard's Villa. This skip works on both Link's Awakening and Link's Awakening DX, but the required inputs slightly differ. This is because pause mechanics differ between the two games, and in Link's Awakening DX it is possible to Start Button Buffer, whereas in Link's Awakening it is not. Below is a video of the skip in full from Link's Awakening DX.
To start, slash the bush over the left hole and jump over it such that Link's body hits the screen boundary. Continue to hold down so that Link will slowly fall into the hole from the bottom side. Your next inputs will differ depending on which game you are playing.
In the original Link's Awakening the next input is frame-perfect. When Link falls a sound effect will play. On that same frame open up the Save+Quit menu and close it. If done correctly, you will see Link standing on the edge of the pit you jumped over. Press down to transition to the next screen.
In Link's Awakening DX you are able to buffer a Start button input during unpause lag. This will advance the game one frame at a time. Keep pause buffering until you hear Link's falling sound effect. If done correctly, you will see Link standing on the edge of the pit you jumped over. Press down to transition to the next screen.
After entering the maze, the inputs are the same regardless of which game you are playing. Follow the maze on screen left and dig in order to obtain the Slime Key. Then move back right and to the bottom right of the bush section. Slash the bush directly to your left and move diagonally into the screen border. If done correctly, you will jump out of the maze.