The easiest way to do this is by turning on the Promiscuous Mode of your network adapter and using a tool such as Wireshark or tcpdump to capture the traffic. This article explains how promiscuous mode works in general and what are some other alternatives to network monitoring. Here is an explanation is somewhat non-technical language (taken from the same blog):
Network traffic on other computers on your local network is inaccessible for you by default. The problem is that all network packets are addressed to somebody and if you are not the specified recipient of the packet, your network adapter must ignore it. But, in practice, from the very emersion of Ethernet most adapters have been able to accept packets even if those were not addressed to them. Just turn on the special – promiscuous – operating mode for your network adapter. In this mode, network adapter accepts all packets flowing within the network segment indiscriminately. In a hub-based network, it would be sufficient to switch the adapter to the promiscuous mode in order to get access to all traffic on the local network, because hub is a primitive device. When it has received a packet from some port, hub simply retransmits it to its other ports. Thus, it would be sufficient to just connect to any port on the hub to monitor Internet traffic passing though the hub.
Further readings: Promiscuous vs Monitor mode.