Triplanar training uses movements in all three planes of motion: sagittal (forward and back), frontal (side to side) and transverse (rotation). Technically all movement has a component in each plane, but here I mean training that deliberately employs the frontal and transverse planes, where most gym training, CrossFit included, rarely goes.
1. It makes you stronger
Muscles attach to bones through tendons, and where they attach determines the directions they produce force. Attachment points are not all sagittally aligned; the human body wasn't built for moving only forwards and backwards. To get the best stretch in your glute you need hip flexion, adduction and internal rotation (a crossover step onto a box with a turned-in toe: try it). Training a muscle in its most lengthened state works its weakest position, which is why the bottom of a squat is harder than the top. Training only sagittally works important muscles through partial ranges, like doing shallow squats forever. Train the full range across planes and recruitment improves, and PRs in the traditional lifts commonly follow.
2. It improves your mobility
Because muscles cross planes of motion, a restriction in one direction usually impacts another. Poor hip adduction, abduction or rotation limits squat depth: if you can't wave your leg left and right well, it can stop you squatting to full depth, because small amounts of those motions are required to drop through parallel. Training all three planes prevents and reverses restrictions in the neglected planes, which improves the sagittal positions you actually care about. And it's more fun than foam rolling.
3. It improves your work capacity
Muscles work in pairs. For a maximal contraction of an agonist, the antagonist has to relax appropriately: you can't max a bicep curl while contracting your tricep. A tight ITB or hip flexor wastes energy on every step of a run and every stroke on the rower, because that tightness must be overcome before any useful force is created. You're a car with one wheel out of alignment, making everything harder for yourself.
4. It carries over to life
Functional fitness is fitness you can use everywhere else: looking over your shoulder to reverse the car, picking your child up on one side and putting them down on the other, rolling over in bed, swinging a racket, side-stepping during the zombie apocalypse. Life is in three planes of motion. Train in the way that most closely replicates it.
Closing thoughts
The traditional sagittal lifts and cardio are excellent. Big squats, deadlifts and presses build maximal strength; running, rowing and cycling build the engine. Complement them with triplanar work and you get strength gains, mobility gains, cardio gains and life gains.