They are (likely) talking about collar stays:
Collar stays (sometimes known as collar sticks, bones, knuckles, tabs,
in the UK, collar stiffeners, and in Eastern Canada collar stiffs) are
Collar stays are smooth, rigid strips of metal (such as brass,
stainless steel, or sterling silver), horn, baleen, mother of pearl,
or plastic, rounded at one end and pointed at the other, inserted into
specially made pockets on the underside of a man's shirt collar to
stabilize the collar's points. The stays ensure that the collar lies
flat against the collarbone, looking crisp and remaining in the
correct place. Often shirts come with plastic stays which may
eventually need to be replaced if they bend; metal replacements don't
have this problem.
However, there is another form of shirt stays:
elastic straps that connect the bottom of a dress shirt to the socks or feet.