I'm trying to write this for everyone, including players with limited technical knowledge of computer connections, and those that aren't really familiar with Digimon World. We've all been there. Running World's battle mode with another player over the web requires some setup, after all.
(Note that this guide is written with Windows 7 and earlier in mind; so far all connection tests with Windows 10 have ended in failure.)
1. Create Identical Systems
For starters, you both need to be running the same version of the game (i.e. two American/NTSC-U copies of Digimon World, or two European/PAL copies, or two Japanese/NTSC-J; the different regions are really different games) and both players need to use the same version of ePSXe. The easiest way to do that is just make sure you're up to date with version 1.9.0. Also ensure that you both have up to date copies of each other's memory card save data (the files that are by default epsxe000.mcr, epsxe001.mcr, etc.) and that both players have each other's data plugged into the same slots in ePSXe. (i.e. if player one's memory card is epsxe000.mcr and it's in slot 2 on ePSXe, it also needs to be in slot 2 for player two's ePSXE.) Both players should also be using identical graphics and sound plugins. (Pete's OpenGL GPU comes as a standard in ePSXe; to play it safe, configure your plugins to the Fast default.) Both players should have a second controller configured, as each one needs to use a different controller in battle mode. (Note that Kaillera always detects the player hosting the game as player 1, and the player that joins them as player 2.) You essentially want your copies of ePSXe and the game to be as close to identical as possible.
ePSXe's cyberpad plugin, which includes the actual client program for linking your systems. (The client is called Cyberpad Kaillera, cpka for short. Hence cpka-v14 is "Cyberpad Kaillera version 1.4." When in use, Cyberpad transmits each player's controller inputs to each other's copies of ePSXe.) When unzipping cpka-v14.zip, extract "kailleraclient" into the ePSXe folder, and move cpka-v12 into ePSXe's "plugin" folder. After that's done, open ePSXe, go to Config > Netplay, select Cyberpad v1.4 and then confirm OK.
2. Connect to a Server
Preliminary testing suggests that for Digimon World, you should have a connection type of at least Good. The quality of your connection in the client is measured in terms of keyframes per second, basically the first and last frames in one animation. This isn't 1988; even Digimon World runs at 40 FPS in battle mode! Setting your connection to Excellent will help avoid things like button inputs being sent late and triggering a pause (and thus a disconnect).
There are around 130 Kaillera servers in all, and the recommended ones for now are those operated by Emuparadise. These have no required connection type, and are general-purpose emulation servers. Many of the others are exclusive to specific games.
(Do not use the pause function while playing with another player. Do not use save states. Do not do anything that would interrupt gameplay, or you risk desynching your games and causing a disconnect.)
3. Set up the Game
roll a three-sided die, and in a best-of-three set having the loser of game 1 choose the stage for game 2, the loser of game 2 choosing the stage for game 3, etc. for larger sets.)
Now grab some friends and go play!