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