It's a shame that the Digimon universe must forever be in the shadow of Pokemon, because had it not been for the latter, Digimon World DS would be a pretty good game. However, because Pokemon set the standard for exciting RPG elements such as wild monster collecting and battling - some years ago, in fact - Digimon World comes off as a game of "meh" proportions.
By far the best part of Digimon World DS is capturing, training and evolving Digimon, those morphing, cuddly-then-monstrous cyborg creatures that aggressively defy laws of evolution that even Pokemon must obey. Digimon are captured in the main game by way of scanning at the beginning of each random battle. Each scan only nets you a certain percentage, but get to 100% or more of a particular Digimon and you can convert them for use in your party or for training. The more you hold out, the higher the scanning percentage, which means more HP and stats for your digital fighter.
You can purchase farms, which you then accessorize with training equipment so the Digimon can level up. This is helpful for strengthening Digimon that aren't currently in your party for use in later, more difficult dungeons, but it's not really necessary. You can just as easily convert higher level monsters later in the game - that way, you get powerful beasts without spending the massive amount of time it takes to train them. So basically, the training is useful only if you plan on collecting as many of the 200 Digimon as you can.