What is minimization (μ-function) in layman tems?
This is minimization in Python:
def mu(p):
    n = 0
    while not p(n):
        n += 1
    return n

It is a slightly more complicated exercise to convert a while to $\mu$, but essentially, $\mu$ performs a search for the first (minimal) number satisfying a given condition, where there is no guarantee that the search will succeed.