First, note the formal difference: etwas ansehen already has an accusative object. Therefore, if another nominal object is to appear, it must be in the dative. (There is only a handful of exceptions to this rule.)
Second, the pattern that builds sattsehen is productive. The adjective is interpreted as a resultative predicative pertaining to the single object, which is in the accusative. The accusative is to be expected in that the object is undergoing a change of state, with the resultant state being described by the adjective. In the case of sich sattsehen, think seeing so much of something that you have become satiated (i.e. don't want to look at it any further). Further examples:
Ich habe mich müde gelaufen.
"I walked so much that I am tired now."
Er hat sich schwarzgeärgert.
Lit. "He was so angry that he is now black.", i.e. "He got really angry."
Sie hat sich warmgeredet.
Lit. "She has talked so much that she is now warmed-up."
Man hat ihn grün und blau geschlagen.
"They hit him until he was black and blue."
Es hat schon wieder jemand eine Katze totgefahren.
Lit. "Someone drove a cat dead again.", i.e. "Someone ran over a cat again."
The predicative doesn't need to be an adjective.
Sie hat sich in Fahrt/Rage geredet. (similar to the third example above)
Note that whether the object is reflexive or not is dictated by plausibility. So although sattsehen is always reflexive, müde laufen does not need to be:
Die Mannschaft hat den Gegner müde gelaufen.
"The team ran their opponent ragged."
Sattsehen likes to appear in the idiom sich (an etwas) nicht sattsehen können.
Die Landschaft war so schön, daß man sich an ihr nicht sattsehen konnte.
"The landscape was so beautiful that you couldn't get enough of it."