Multiplication is nothing but repeated addition. We know that 0+0=0.
So multiplying a number with 0 means adding 0 that much number of times .This means that nothing is added. The value remains zero.
If you need a theoretical proof, then it may be as follows,
0+0=0
Multiplying both sides by a, which is any number
a*(0+0)=a*0
a*0 + a*0 =a*0
a*0=0
Whatever may be the number, whatever times it may be added ,the result remains zero.
We must be proud to be Indians ,who discovered this magic number