In particular, the constants are very critical to the algorithm's security. If the constants are maliciously chosen, the algorithm could be "backdoored" with an exploitable structure. For example, if you modify just four 32-bit constants in SHA-1, without touching any other operations, it becomes nearly trivial to obtain collisions (current SHA-1 collisions have required enormous amounts of compute to achieve): https://malicioussha1.github.io/

The constants for SHA-256 are chosen to be simple mathematical constants, reducing the number of "degrees of freedom" for an attacker to manipulate the standard. At the same time, the constants need to be "random" enough to avoid producing exploitable structure. It's a careful balancing act! Roots of primes and functions or digits of transcendental numbers (e, pi, etc) make good random-ish numbers which don't provide much room for manipulation.