Here's the cryptography theory perspective.
We want block ciphers to resemble pseudo-random permutations (PRPs). PRPs are a desirable modeling goal because a block cipher under a given key is a permutation on the input, and a PRP is simply a random collection of permutations. The block cipher's key can never be better at creating permutations than an actual random sampling of them, but we want it to be as close as possible. Detectable deviation from PRP-like behavior is considered a weakness in a block cipher.
To compare a block cipher to a PRP we use the CPA model, where an attacker queries a black box with plaintext and receives the corresponding permuted output. They try to determine if the black box is choosing the ciphertext output by apply a random permutation or the given block cipher with an unknown key to the plaintext. If they guess correctly they win the game. If they can win the game with probability greater than 50% then they've broken the block cipher. (Look at these notes, pages 1 - 8, specifically 7, for a picture and more precise definition of this model.)
An attacker can distinguish a block cipher from a PRP if said block cipher has linear properties. They can learn those linear properties then query the black box with plaintexts that should produce certain properties in the ciphertext. If the black box replies back with ciphertext that matches those expected properties for more than 50% of the input queries, then the attacker guesses that the black box houses the block cipher because a random permutation would honor those linear properties with only 50% probability. (Otherwise, they guess that the black box is using a random permutation.) Then the attacker wins the distinguishing game.
Of course, if you're looking for a practical reason, this raises the question of why we care about PRP models to begin with.