The Automated Ground Umbilical Systems (AGUS) Project contains a good overview of umbilical systems.
The fluid lines are separated by "quick disconnects". Here is some information on them from that paper.
Quick disconnects (QD) provide fluid servicing
either directly to the vehicle or to a mobile facility.
Considerable effort has been made in developing QDs
for various types of applications . They can range
from sizes of 1⁄4 in. to 8 in. and can be of the latching
or non-latching type. Generally non-latching QDs are
used for application where separate locking
mechanisms are used for mating umbilical plates
together. Figure 18 shows a bellows type QD that uses
the bellows preload and pressure for sealing a spherical
mating surface. The bellows is a special formed section
that acts as a spring. These types of QDs are typically
used for T-0 umiblicals to minimize alignment issues
during disconnect. Additionally, they may have
poppets spring loaded on the groundside and/or flight
side to isolate systems after separation. A single
ground side poppet is shown.
A second type of QD uses a slip on design where
seals are used between the circumferential surfaces
(Figure 19). This type of design minimizes QD
interface loads from pressure and preloads but requires
a longer engagement.
(added emphasis on poppets that seal the connector after separation)
The NASA paper from the Space Vehicle Design Criteria (Chemical Propulsion) series Liquid rocket disconnects, couplings, fittings, fixed joints, and seals goes into a great deal of detail if you wish further information. See chapter 2.1 Disconnects.
As mentioned above, poppet valves are typically used to shut off the fluid lines after disconnection. More rarely, ball valves are used.
This paper gives examples of externally and internally operated poppet valves.