Why do block ciphers need a non-linear component (like an S-box)? Why is there a requirement of "Non-Linear functions" as a component of many popular block ciphers (e.g. the S-box in DES or 3DES)?

How does it make the cipher more secure?

The only intuition I have is a non linear function can have many roots (solutions). E.g. for a non linear $y = F(x)$, if attacker knows $y$ to find out the $x$, there might be many choices which satisfy $y= F(x)$ hence it adds up complexity at attacker's end when he tries to make his way back (trace back) from cipher text to plain text.

Any references or detailed material are appreciated.
If a block cipher is linear with respect to some field, then, given a few known plaintext-ciphertext pairs, it is possible to recover the key using a simple Gaussian elimination. This clearly contradicts the security properties one expects from a secure block cipher.