The name Trace here is related to the notion of the trace of function in general which basically means the restriction.
I other word, the Trace of a function in general its restriction (whenever it makes sense) on a subset of ITS DOMAIN ( usually named as trace set of the original function) .
But be aware the actual situation is much more delicate. Since given a function defined almost everywhere on a bounded domain it is not always possible to take its TRACE (or its restriction ) on the boundary.
Rather if we consider a CONTINUOUS function up the the boundary , it is possible to take its TRACE (or its restriction) on the boundary. This means that the OPERATOR that maps a continuous function up the boundary to it TRACE (to its restored) on the boundary is WELL define . Moreover, this OPERATOR is bounded if we endowe both depature and arrived spaces with suitable L.p. or Sobolev NORMS.
THIS THEN GIVES THE NAMES TRACE OPERATOR.
FINALLY since the space of continuous function up to boundary is dense IN SOME SUITABLE L.p. space or SOBOLEV SPACE, we know that, OUR LATEST TRACE OPERATOR CAN BE CONTINUOUSLY EXTEND TO THE corresponding L.p. or Sobolev spaces.
AND IT PRESERVES THE NAME TRACE OPERATOR AS WELL.
1. There are some cases where the density argument may not apply but One can still use HANH-BANACH theorem to get A CONTINUOUS EXTENTION.
2. One may chose other spaces instead of the space of continuous function up the boundary. Like the space consisting of functions which are restrictions of functions that are smooth (on a bigger domian ) on closure of the DOMIAN (the we are working with ).