What does it mean when the white spots from tonsillitis starts to turn black after starting an antibiotic regimen?

The infection is: likely responding to the antibiotics and the pus is turning color. Necrotic tissue can also give the darker/ blackish appearance which would indicate failure of therapy with antibiotics. Rely on other signs (i.e. worsening pain, worsening fever, weakness, increased phlegm or shortness of breath, etc.) to give you an idea if things are better or worse.