Does the moon really turn black during a total solar eclipse?

No. If the corona of the sun were not so bright, you would see the moon very faintly illuminated by earthshine.  Our Earth is fully illuminated during the eclipse and it reflects quite a bit of light into space. Some of this lands on the lunar surface and provides a secondary source of illumination. But because the sun’s corona is so bright, your eyes will not see this earthshine effect.