There is a connection that's not very obvious in those diagrams, from L3 terminal (third along the upper side) to the contactor coil feed.
Operation-wise, you can think of it as there being two parts to the circuit.
The first is a standard "latching relay" circuit with the buttons, contactor coil and one contact.
Power feeds through the normally closed "stop" button to the start button and a relay (or contactor) contact.
The other side of of both the start button and contact connect to the relay coil.
When the start button is pressed, power is connected through to the coil and the contactor/relay pulls in - closing the contact across the start switch, so power stays on when the button is releases.
Pressing the stop switch disconnects power from the contact/coil and the contactor or relay drops out.
This shows a basic latching relay circuit, functionally the same as your diagrams; the circle represents the relay / contactor coil.
The other part is the load circuit. That just switches power through separate contacts on the relay/contactor to whatever load is being operated.
[Edit - typo].