Inverting the encryption equation of a symmetric cipher
Addition modulo $N$ in the above example is another way of key mixing that adds more non-linearity to the cipher as it operates across a group of $N$ bits, vs XOR which is addition modulo 2 on individual bits. For example, where $N=2^6$ or 64:

$44 ⊞ 44 = 24$

$44 ⊕ 44 = 0$

Addition within larger groups is also used in mixing operations such as Pseudo Hadamard Transformations. Key addition instead of XOR also plays a prominent role in the block ciphers Twofish and IDEA.

Edit:
Here are posts with more detail on the choice of addition:

What exactly is addition modulo $2^{32}$ in cryptography?

Why Addition Mod 32?