Po appeared in right place at the right moment. Just when Oogway was choosing the Dragon Warrior, he accidentally pointed at Po (instead of Tigress?). As Oogway later tells Shifu there are no accidents so appearance of Po must be planed/influenced by some other forces.
Shifu: The panda? Master, that panda is not the Dragon Warrior. He wasn't even meant to be here. It was an accident.
Oogway: There are no accidents.
Shifu: [Sighs] Yes, I know. You said that already... twice.
Oogway: Well, that was no accident either.
Shifu: ... Thrice.
As Oogway truly believed in this "accident" he sticks with his decision.
To understand Oogway's philosophy we need to consider another moment from the movie when he talks to Shifu:
Oogway: My friend, the panda will never fulfill his destiny, nor you yours, until you let go of the illusion of control.
[points at peach tree]
Look at this tree, Shifu. I cannot make it blossom when it suits me, nor make it bear fruit before its time.
Shifu: But there are things we can control.
[kicks the tree so that peaches fall]
I can control when the fruit will fall!
[he slices a peach and throws the pit to the ground]
I can control where to plant the seed! That is no illusion, Master!
Oogway: Ah, yes. But no matter what you do, that seed will grow to be a peach tree. You may wish for an apple or an orange, but you will get a peach.
Shifu: But a peach cannot defeat Tai Lung!
Oogway: [folding dirt over the peach pit] Maybe it can, if you are willing to guide it, to nurture it. To believe in it.
Shifu: But how? How? I need your help, master!
Oogway: No, you just need to believe. Promise me, Shifu. Promise me you will believe.
Shifu: I... I will try.
In both these dialogs Oogway presents his philosophy that is similar to what Aristotle said:
Aristotle made a distinction between the essential and accidental properties of a thing. For example, a chair can be made of wood or metal, but this is accidental to its being a chair: that is, it is still a chair regardless of the material from which it is made. To put this in technical terms, an accident is a property which has no necessary connection to the essence of the thing being described.
Po is a panda, it's his essence. But he could be shaped into the Dragon Warrior - accidental property.
According to Aristotle, there is always a reason for everything that happens. Your experiences are designed to shape you, define you and, hopefully, grow you into the mightiest you possible.