Because "Osman" is an Arabic name, and in Arabic it's actually written and pronounced "Othman" or "Uthman". The "th" kind of sounds like the "th" at the end of "tooth".

Turkish and Persian don't have that sound, so they replace "th" with "s" and it becomes Usman or Osman. This is the pronunciation you'll find wherever Persian influence existed, for example, the Indian subcontinent. Probably more people say "Osman" rather than "Othman" worldwide, which is why many wonder why it's not the Osmani Empire.

Nevertheless, the name is Arabic and in Arabic it's pronounced "Othman". As for how Othman became Ottoman, I believe it's the Italian spelling borrowed through French to English.

There is another theory that Osman Ghazi's original Turkish name was Ataman, and perhaps Ottoman is a corruption of that. In support of this theory, the earliest Turkish mentions of his name spell it "Otman" (with the "toi" sound like a soft "t"), and only later is it spelled "Othman". Meaning, perhaps he Arabized it in later life.

That may or may not be true. However, it's not a nnecessary explanation, you can understand the sound change simply on the basis of transcribing Arabic to Turkish or Persian, with the English form being closer to the Arabic.

The name is "Osmanlı" in Turkish, which would translate as 'Osman's'. Its English name derived from the French Ottoman or Italian Ottomano both nations having contact with the empire earlier than the British. That word itself comes from the Arabic 'Uthman'. This is the traditional spelling of that name in Latin languages, chosen to correspond to its phonetics. (e.g Uthman ibn Affan, Islamic Khalifa of 7th century)