First I would like to give some more details about what is involved in obtaining scattering amplitudes in string theory and the specific one for the scattering of four open string tachyons.

Generally, for calculating string scattering amplitudes, one starts by drawing the string diagrams (worldsheets) that correspond to the process one is interested in, as Riemann surfaces in the complex plain and brings them by means of conformal transformations into a convenient standard representation. The scattering amplitude of the process is then obtained by integrating over the moduli space that parameterizes all worldsheets with the same initial and outgoing state.

The worldsheet for the scattering of 4 open string tachyons can for example be considered as a disk in the compex plain with four punctures at the locations $P_1$, $P_2$, $P_3$, $P_4$, where the incoming and outgoing particles are inserted. By a conformal transformation, this worldsheet can be mapped to the upper half plain such that the points of insertion appear on the real line

\(P_1 \rightarrow x_1 = 0\)

\(P_2 \rightarrow x_x = \lambda\)

\(P_3 \rightarrow x_3 = 1\)

\(P_4 \rightarrow x_4 = \infty\)

To define the conformal (or linear fractional) transformation only 3 points are needed, $\lambda$ is left as a free parameter and its range $0 < \lambda < 1$ therefore corresponds to the moduli space of the $2D$ Riemann surfaces with 4 punctures to be integrated over to obtain the scattering amplitude of the process.

The Ansatz for the scattering amplitude depends on the momenta of the 4 incoming and outgoing particles $p_i$ and can be written as

\(A(p_1,p_2,p_3,p_4) = g_0^2\int d\mu\)

It is proportional to the coupling constant $g_0$ squared because two interaction vertices are involved, and it contains $d\lambda$ to integrate over the moduli space of the relevant worldsheets. Considering momentum conservation and the on-shell condition of the incoming and outgoing particles, and taking into account that the integration measure has to be conformal invariant, the scattering amplitude can be rewritten as ($\alpha'$ is proportional to the string length squared in natural units)

\(A(p_1,p_2,p_3,p_4) = g_0^2\int\limits_0^1 d\lambda\lambda^{2\alpha'(p_1,p_2)}(1-\lambda)^{2\alpha'(p_2,p_3)}\)

Transforming to the Mandelstem variables $s$, $t$, and $u$, and defining the expressions

\(\alpha(s) = \alpha's +1\)

\(\alpha(t) = \alpha' t +1\)

the Veneziano amplitude can finally be written as

\(A(p_1,p_2,p_3,p_4) = g_0^2\int\limits_0^1d\lambda \lambda^{-\alpha(s)-1}(1-\lambda)^{-\alpha(t)-1}\)

which exactly corresponds to the definition of the Euler beta function as given in the Wikipedia link in the question for example. The symmetry of this amplitude in exchanging s and t (I think it is called channel duality?) is a specific stringy feature.

Somewhat handwavingly, to model the strong interaction by strings mesons are viewed as a pair of quarks sitting at the end of a string, that is represented by a thin tube of color flux lines. The confinement of the strong interaction can then be explained that the fact that when trying to separate the two quarks more and more mesons are produced such that the "binding energy" linearly increases with the separation $E_B = T L$ and $T$ is the string tension.

In addition, it has been observed for hadrons when plotting the angular momentum against the energy (or mass) squared, they appear as points on lines $J = \alpha' E^2$ which can be modelled by a classical rotating string. The discrete mass spectrum (the points on the line) of the hadrons including the offset of the linear relationship can be explained by a quantized open rotating string.