How does a single Wireless LAN Controller (WLC) connect one wired network to many WLANs and VLANs?

Its physical ports act as trunk ports carrying many VLANs (Virtual Local Area Networks), and it uses software-created virtual interfaces — just like a switch's VLAN interfaces — to map each WLAN (Wireless Local Area Network) onto a VLAN. One port serves many APs (Access Points), and each AP can broadcast many WLANs.

* Trunk port carries many VLANs; each WLAN maps to a VLAN interface. *

This is the piece that lets one controller serve a whole campus from a few cables:

• Layer 2 switch ports + virtual interfaces — the WLC has physical ports plus interfaces created in software that behave very much like VLAN interfaces on a switch.
• Ports are trunks — each physical port is essentially a trunk that carries traffic for multiple VLANs down to a switch, which distributes it to the APs.
• Many-to-many — each physical port can support many APs and WLANs, and each AP can broadcast multiple WLANs at once.

That's why every WLAN you create is mapped to an interface (a VLAN): the mapping is what puts each group of wireless clients onto the right subnet.

Go deeper:

• Virtual LAN (Wikipedia) — VLANs and trunk links (one link carrying many tagged VLANs), the model the WLC's ports and interfaces follow.