Why does using a prime-order subgroup in DLP improve security? Let's consider a discrete logarithm


  $\beta \equiv \alpha ^{x} \bmod \,\, p$


We can solve it using Pohlig-Hellman algorithm. And, if $p-1 = tq$ where $q$ is a large prime factor, we can avoid any leakages by choosing $\beta=a^{t}$.

This is what my professor said at lesson, however i don't succeed in understanding why it avoids any leakages. Can you explain to me why ? Also with calculations if it's possible
Probably what your professor meant is that you start with any group element $\alpha$, and then use $g := \alpha^t$ as the generator for a cryptosystem such as Schnorr signatures, as long as $g$ is not itself the identity.

Why?  If $g \ne 1$, then $g$ is guaranteed to have prime order $q$, because $g^q = (\alpha^t)^q = \alpha^{tq} = \alpha^{\phi(p)} = 1$, and since $q$ is prime there are no smaller orders possible for $g$ (just start over with a different $\alpha$ if it is).

Then when you choose a public key $\beta := g^x$ for secret $x$, you're guaranteed that $\log_\alpha \beta \equiv 0 \pmod t$.  (Of course, you could have equivalently chosen $\beta := \alpha^{tx}$, but precomputing $g$ may be cheaper.)

Now since the group generated by $g$ has prime order, Pohlig–Hellman has no advantage over any other DLOG algorithms.