A favicon is the small square icon that sits on a browser tab next to the page title, in a bookmarks list, and on the home screen when somebody saves your site to their phone. Short for favourite icon, from the days when bookmarks were called favourites.
Why a missing one is noticed
Nobody has ever rung a firm because of a good favicon. Plenty of people have quietly closed a tab that showed a blank page icon while the three tabs beside it showed proper logos.
It is a small signal of whether anyone finished the job. A homeowner comparing three quotes on three tabs sees yours as the unlabelled one, and that is the impression you are arguing against.
What you actually need
Two or three PNG files does it. A 32 pixel square for the browser tab, a 192 pixel square for Android home screens, and a 180 pixel apple touch icon for iPhone. Modern browsers are happy with PNG, so the old .ico format is optional.
Design it as a shape, not a shrunken logo. Your full business name will be an illegible smudge at 32 pixels. One letter, a mark, or a simple symbol in your brand colour reads at that size.
A worked example
A fencing contractor sends a quote by email with the website link in it. The customer opens it, keeps the tab open all week while they think about it, and has four other tabs going.
On Friday they go back to the one they can find. A yellow square with a clear letter on it is findable. A blank sheet of paper icon is not.