As of writing, LAN multiplayer in Fungicide only works for the Killing Spree mode, and supports two players. Have more than one friend? That's too bad, at least for right now.
Connecting on Fungicide for local multiplayer is a simple process. You select "New Game" then "Killing Spree."
When prompted for the number of players, select two. It will then ask you if you want to host a game or not. One of you needs to host the game, it doesn't really matter which. Once the game is created, the host's local IP will be displayed on his screen.
Player two needs to say no when they are asked to host. Then, you're going to be asked for the host's IP. Enter the one displayed on the host's screen. Once that's done, you should both be in the game lobby that allows the host to edit settings.