What is Extreme Learning Machine? Why tuning of weights is not required?
ELM is actually solving a generalized linear problem :

\begin{equation}
H\beta = Y 
\end{equation}

where $H$ is the matrix of the hidden layer output, $\beta$ is the weight vector (or matrix) connecting the hidden layer and the output layer and $Y$ symbolizes the target of your training. 

The goal is to find $\beta$ using the pseudo-inverse of H.

Solving this problem is equivalent of finding the $\beta$ achieving the minimum of the Least-square problem $||H\beta - T ||$ : 
\begin{equation}
\beta = \underset{\beta}{\text{min}} \left \vert\left \vert H\beta - T \right\vert\right\vert 
\end{equation}
To answer your question 


  How is it possible to assign values to links without actually seeing the Training set?


Since $H$ depends on the activation functions, on the initialized weight matrix linking Input layer and hidden layer and on each hidden node bias, it encompasses all the input information and ensures a learning by tuning $\beta$ coherently.

You can find lot more details in ELM Theory and Application or in this ELM presentation for example.