Could you explain to me what happens from the linguist’s point of view when the sounds meet in the speech?
Briefly (because stuff like this happens whenever words meet up in speech,
which is to say in every sentence), the phonemics of words that
(occurring in a phrase, where words is stressed and that is unstressed)
is something like:
The big problem is that long consonant cluster in the middle:
The /r/ just colors the preceding schwa vowel to [ɚ], so it can be ignored.
But the /dzð/ cluster starts with a dental stop and transitions into a postdental sibilant /z/,
and then an interdental fricative /ð/, all of which requires a lot of complicated lip
and tongue movement, with breath coordination.
So what happens in practice is that things get lost. Consonant clusters are regularly simplified;
sixths changes from canonical /sɪksθs/ to /sɪkss/, for instance.
In the case of words that, the final /ð/ usually drops, producing
which sounds like words it or words at, but is understood as words that at ordinary speech rates.