The intention is very important for the fast, it’s actually fardh. Without the niyyah the fast will not be accepted. You don’t have to verbally say that you intend to fast, but you should make intention in your heart to fast the previous night which can be as late as right before Fajr time.
Sunan Al Nasai, Book of Fasting, Hadith 2331.
Also, check out this video from around the 9 min. mark to the 13 min. mark and then from around 15 min. mark to the 19 min. mark, to hear about the niyyah or intention of fasting.