I wondered this before myself, I found that [Wikipedia had most of my answers][1]. 

> The Forgotten Realms is a fantasy world setting, described as a world
> of strange lands, dangerous creatures, and mighty deities, where magic
> and supernatural phenomena are quite real. The premise is that, long
> ago, the Earth and the world of the Forgotten Realms were more closely
> connected. As time passed, the inhabitants of planet Earth have mostly
> forgotten about the existence of that other world – hence the name
> Forgotten Realms.

The AD&D 2nd edition *Player’s Guide to the Forgotten Realms Campaign* (TSR2142) explains on page 2 the name is based on its relation to our own world losing contact with and forgetting it:

> ### What are the Forgotten Realms?
>
>Some theorists explain dragons and other fantastic occurrences by postulating parallel worlds. In times past, they believe, travel between mundane worlds — like our own — and more exotic locales was easy and frequent. But we have lost contact, and lost the worlds themselves. Lost, or forgotten.
>
>Abeir-Toril, more commonly Toril, is an Earth-sized planet dominated by a continent in its northern hemisphere. Called Faerûn in the west, it is here that the Forgotten Realms lie. […]


  [1]: https://en.wikipedia.org/wiki/Forgotten_Realms