[tromp]

summary:

    -1*-1 =
    -1*-1 + -1*1 + 1 =
    -1*(-1 + 1) + 1 =
    -1*0 + 1 =
    1

[mathnmusic]

The explanation that appeals to me:

If number N is an arrow on the number line from 0 to N, then multiplying N by -1 flips the arrow with the result -N. Multiplying by -1 again would be another flip, taking you back to N. So a flip followed by a flip is same as no change (i.e. multiplicative identity 1).

[mantap]

This would be clearer if the -1 was in parentheses (I can't do it because I'm on mobile). I think you are missing a plus sign on line two.

[nabla9]

in complex plane you can go

e^[i(2nπ+π)] × e^[i(2nπ+π)] =

...

e^[2i(2πn+π)] = 1