I was not able to prove this by myself and also have not found any proof online. Since it is often stated as a fact, I assume it should not be a difficult statement to prove.
The definition of $T_{p,q}$ in this case is as follows:
Torus links are produced by choosing a pair of integers $p$ and $q$, with $p$ positive; forming a cylinder with $p$ strings running along it, twisting it up through “$q/p$ full twists” (the sign of $q$ determines the direction of twist) and gluing its ends together to form an unknotted torus in $\mathbb{R}^3$. The torus is irrelevant — one is only interested in the resulting link $T_{p,q}$ formed from the strands drawn on its surface.
This definition is taken from Roberts - Knotes.
I would appreciate any kind of help.
Do you see why there are $\gcd(p,q)$ strands? Look at the base at the bottom of the cylinder, where the strands are just dots, and number the dots from $0$ to $p-1$. If you take the $k$th dot and follow the strand until you reach the top of the cylinder, you come back at the bottom of the cylinder but at the $(k+q)$th dot. So we see there are $\gcd(p,q)$ strands.
We can rotate a strand on the cylinder around the axis of the cylinder (so that it stays on the cylinder) until we recover another strand. The surface covered in this way is $I × S^1$. This is one of the components. We can repeat this process $\gcd(p,q)$ times until we get back our original strand, and it gives the $\gcd(p,q)$ components of the torus.