Tensor index in special relativity?

Here $Lambda^mu{}_alphaLambda_mu{}^beta$ represent matrix multiplication. But in matrix multiplication elements in a row should be multiplied with elements in a column, and added. Here the index notation represent, each element of a column of first matrix is multiplied with the corresponding elements in another column of the second matrix, and added, means two matrix are multiplied by taking transpose of first matrix. The transpose of first matrix is $(Lambda^mu{}_alpha)^T= Lambda_alpha{}^mu.$ Then it become by index cancellation $Lambda_alpha{}^muLambda_mu{}^beta=g_alpha{}^beta.$ Thus it is obvious that $ alpha$ represent row and $beta $ represent column.

You can verify these formulae satisfy $Lambda_mu^{:nu}=eta_{murho}eta^{nusigma}Lambda_{:sigma}^rho$, so the equation you ask about can be rewritten as $delta_alpha^beta=Lambda^mu_{:alpha}eta_{murho}eta^{betasigma}Lambda^{:rho}_sigma$, which uses only the first version of the Lorentz transformation. You could similarly switch to only using the second viz. $Lambda^mu_{:nu}=eta^{murho}eta_{nusigma}Lambda_rho^{:sigma}$. The two conversions are equivalent because$$eta^{murho}eta_{nusigma}Lambda^{:sigma}_rho=eta^{murho}eta_{nusigma}eta_{rhokappa}eta^{sigmalambda}Lambda_{:lambda}^kappa=delta^mu_kappaeta_nu^lambdaLambda_{:lambda}^kappa=Lambda_{:nu}^mu.$$