Unlike most champion abities that when ranked up increase in Mana cost, putting Ranks in Udyr's stances LOWERS their Mana cost. And, since technically Bear stance is the most used on Udyr (E....e......e......e.....are you where you need to be now? If no, e more. If yes, push other buttons) So leveling his Bear and Tiger stance, arguably the 2 most used stances, he is decreasing his own Mana costs. He's able to use Bear stance more frequently outside of combat and camps. He can change stances faster in combat w/o relying on Monkey Stance to help mitigate 2-3 seconds of stance changing (this saves a bit of Mana)
Plus, the stun duration if increased. I believe? Ideally you approach your enemy with Turtle Stance active, and press e to rush them with Bear Stance. Best case senario, by the time you are able to land a Bear blow, your stances are on cooldown and you can immediately activate Tiger stance. If you rapid press r for Phoenix stance you'll still get in the initial Tiger proc and at least 2 blows before Phoenix stance kicks in with max Monkey Stance stacks and Tiger stances atk speed steroid active.
If they aren't dead yet, just switch to bear stance/flash/punch kittens more. By not sinking points into Bear stance, an Udyr player that isn't careful may accidently run out of mana running from camp to camp, they go to gank, and into their 8th stance they've run Mana out and can't continue to meaningfully engage and have to retreat. Most likely to recall and if they go for their blue, they are weak and vulnerable anyway.
By buying lifestea items, a player can circumvent the Lifesteal bonuses given by Turtle stance. So when they do use Turtle stance it's not bc they needed the lifesteal, but it helps anyway. Also the Sheild strength scales off of AP. So turtle stance is most effective when Udyr has purchased some AP. Triforce for example. Or Hourglass. Don't hate.