A bit of detail:

In Excel 2002, I copy a series of zeros from a cell formatted as Text.

Paste places the zeros in the destination cell and changes its format
to Text. If I then enter a series of zeros in that cell, they all
show.

Paste Special - Values places the zeros in the destination cell and
keeps its format as General. (!?!) I see all the zeros. But if I
enter a series in zeros in that cell - or even activate the cell for
editing but making no changes - I only see a single zero after pressing
Enter.

If you have an earlier version of Excel, it may be inserting the
apostrophe because Paste Special - Values keeps the destination cell's
formatting intact and your destination B6 is in General format. And
the Microsoft programmers hadn't yet figured out how to hide the
apostrophe.