does that sobie thing refer to the subject?
Exactly. Siebie and sobie (and sobą too) are different cases of the same word. They are used when the subjects do action on themselves.
Here's the declension:
M. (none, because object is never in nominative in Polish)
D. siebie (or się)
C. sobie
B. siebie (or się)
N. (z, ze) sobą
Mc. (o, na, w...) sobie
W. (none, as for nominative)
examples:
D.
Wysłał list do siebie.
He sent a letter to himself.
C.
Obciął sobie palec.
He cut his finger off.
B.
Rozpoznał siebie w lustrze.
He recognised himself in the mirror.
N.
Rozmawiał sam ze sobą.
He talked to himself.
Mc.
Opowiedział o sobie.
The told about himself.
EDIT:
In D and B siebie is used when emphasized. Otherwise się is more natural.
But if a preposition is put before D then it must be siebie (not się) - as in the example above.
Cinek