Right, so for myself and all the other idiots out there (of whom I am the apparent appointed leader)…
Let's look at Bob the fighter. He is a crafty sort, having taken the whole spring attack feat tree, and he likes to use it. Bob has also gotten himself some nifty magical items, which boost his speed to 40' / round!
Today, Bob is out adventuring and comes across Humlug the troll. Bob naturally sends his paladin friend and his barbarian friend to fight in the 'front line' while he himself takes advantage of his wonderful feats/feet.
Here is a poorly drawn map…
So, Bob is the "B" square, and wants to attack the troll. The yellow area is the troll's reach. As you can see, Bob has two (well, infinite, but let's just look at these two) paths to take, red and blue.
First, let's look at the red path. Bob can use his spring attack feat to move up to the troll and attack from square 3F, then tumble through to square 3C. His total movement for the round is 40' and he must make a tumble check to pass through the trolls square. Notice that squares 3E and 3D cost twice as much movement, because Bob is tumbling.
Now, let's look at the blue path. Bob can use his spring attack feat to move to 3F and attack, then move again to 3C. This requires no tumble check, and draws no attacks from opportunity from the troll. The total movement for the blue path is 35'. Notice that square 3D costs twice as much, because it is the second diagonal Bob has moved in this round.
It should be noted that if there are other enemies, they will get attacks of opportunity as normal. The spring attack feat only denies the troll attacks of opportunity.
Ok, I think that's it. Good luck!
-Tatsu