Explain mixing bijection with an example
What they are referring to as a mixing bijection is the MixColumns step of the AES round function. It functions to improve diffusion by reversibly scrambling the state of each column. Recall that the block state consists of a 4x4 matrix of byte-sized elements (totaling 16 bytes, or 128 bits), so each of the four columns are composed of four vertical elements. The MixColumns transformation is applied to each column in parallel. It is just one step of several. Together with AddRoundKey, ShiftRows, and SubBytes, it makes up a single AES round. Ten to fourteen rounds are applied to the input for a single block.

It is referred to as a mixing bijection instead of just a bijection to imply that the output is random. Not all bijections appear random of course. This isn't standard terminology, which explains the confusion. Just think of it as a bijection whose output appears fairly random, making it useful for diffusion.

I strongly recommend you read A Stick Figure Guide to the Advanced Encryption Standard (AES). It is a wonderful explanation of the intrinsics of AES, including MixColumns, and is very easy to understand.