Why is a number of shares that is greater than a threshold required to construct secret?
To show that the adversary could not retrieve the secret, you can show that two different secrets will implies the same shares, and then an adversary that has access to these shares could not retrieve the secret.

To show that the adversary has no bit of information about the secret when he has $t'$ strictly less than $t$ shares, you can show that if you take two different $K_1$ and $K_2$, and compute the cardinality of $a$'s (for $k_1$ and for $K_2$) such that you obtain the $t'$ shares. Because they are the same, and because $a$ is uniformly picked at random, you deduce the perfect secrecy.
First I'm claiming that $a$ is chosen unifromly at random in $\{0, \dots, M-1 \}$ (I can give you more details, if you aren't not convinced.
Let $\left(y_1, \dots, y_{n-1}\right)$ a vector of shares.\newline
We call $\psi$ the isomorphism given by the chinese remainder theorem from $\mathbb{Z}_{m_1} \times \dots \times \mathbb{Z}_{m_n} \times \mathbb{Z}_{p}$ to $\mathbb{Z}_{Mp}$, and $\phi$ the invert isomorphism.

$$Card \{a \in \{0, \dots, M-1 \} / \exists \left(z, z'\right) \in  \mathbb{Z}_{m_n} \times \mathbb{Z}_{p}, \phi \left(K_1 + a*p\right) = \left(y_1, \dots, y_{n-1}, z, z'\right)\}$$

$$ = Card \{a \in \{0, \dots, M-1 \} / \exists \left(z, z'\right) \in  \mathbb{Z}_{m_n} \times \mathbb{Z}_{p}, \phi \left(K_1\right) + \phi\left(a\right)*\phi\left(p\right) = \left(y_1, \dots, y_{n-1}, z, z'\right) \}$$

$$ = Card \{a \in \{0, \dots, M-1 \} / \exists \left(z, z'\right) \in  \mathbb{Z}_{m_n} \times \mathbb{Z}_{p},  \phi\left(a\right)*\phi\left(p\right) = \left(y_1 - \left(K_1 \mod m_{1} \right), \dots, y_{n-1} - \left(K_1 \mod m_{n-1}\right), z - \left(K_1 \mod m_{n}\right), z' - \left(K_1 \mod p\right) \right) \}$$

Because $p \mod p = 0$, we deduce, that necessarily  $z' = K_1 \mod p$ (and such $z'$ exists).

$$ = Card \{a \in \{0, \dots, M-1 \} / \exists z \in  \mathbb{Z}_{m_n},  \phi\left(a\right) = \left(\frac{y_1 - \left(K_1 \mod m_{1} \right)}{p \mod m_{n-1}}, \dots, \frac{y_{n-1} - \left(K_1 \mod m_{n-1}\right)}{p \mod m_{n-1}}, \frac{\left(z - \left(K_1 \mod m_{n}\right)\right)}{p \mod m_n}, 0\right) \}$$

$$ = Card \{a \in \{0, \dots, M-1 \} / \exists z \in  \mathbb{Z}_{m_n},  a = \psi\left(\frac{y_1 - \left(K_1 \mod m_{1} \right)}{p \mod m_{n-1}}, \dots, \frac{y_{n-1} - \left(K_1 \mod m_{n-1}\right)}{p \mod m_{n-1}}, \frac{\left(z - \left(K_1 \mod m_{n}\right)\right)}{p \mod m_n}, 0\right) \}$$.

This cardinal doesn't depend of $K_1$ (it's equal to $m_n$).