Connections to the nodes provide access to the voltages available.
Voltage ladders are useful for providing a set of successive voltage references, for instance for a flash analog-to-digital converter.
A voltage drop occurs across each resistor in the network causing each successive "rung" of the ladder (each node of the circuit) to have a higher voltage than the previous one.
The voltage referenced to ground at any node is simply the sum of the voltages dropped by each resistor between that node and ground.
The voltage of a node referenced to ground is the sum of the drops across all the resistors, but it's now easier to consider all these resistors as a single equivalent resistance RT, which is simply the sum of all the resistances between the node and ground, so the node voltage is given by V×RT/Req.