Why was it so important to the US that they have access to Eastern Europe?

For the same reason it was so important to Russia to invade the same area and take so much land.

Russia had suffered invasion after invasion, and each time they took more land when they fought back as a buffer against further invasions.
The U.S.A. does the same thing, but not as an invasion force, just as a political and economic presence to protect U.S.A. interests.

Eastern Europe has, for many decades, been closed in many ways to United States efforts to spread their political and economic influence, and the sudden gap in the wake of fall of the U.S.S.R. has left everyone scrambling, not just the U.S.A.

Historically it was important to the UK, but for the same reason it is today important to the US: They want to exploit their resourses of potential wealth.