What does Eliza's act of fetching Higgins's slippers symbolize in Pygmalion?

It symbolizes her loyalty and devotion to Prof. Higgins.
Higgins has successfully trained Eliza to be a lady of quality in much the same way that you'd train a puppy to fetch your slippers. Despite her outward demeanor as a duchess, Eliza is still instinctively deferential towards her alleged social superior and so doesn't hesitate to act like Higgins's servant.
But Higgins doesn't show the slightest hint of appreciation for anything Eliza does for him. That includes her astonishing transition from flower-selling guttersnipe to belle of the ball. As far as Higgins is concerned, he and he alone deserves all the credit for that.
No wonder, then, that Eliza angrily throws Higgins's slippers right at him. Her devotion and loyalty have not been acknowledged. From now on, she's determined to be more assertive, less deferential.