I'm not sure there is a "laymans" terms explanation for this. A single & is a bitwise and. This means it will go through the left and right, compare them bit by bit and the result is determined by the and function:

src1   src2   dest
0      0      0
0      1      0
1      0      0
1      1      1
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In your example src1 is $mode, src2 is 07777 and dest is $perms. This is mostly used to zero out bits, or mask off portions of bits.
The bitwise or operator, |, is used to set bits in a bit-string.
Hope this helped.