Let's see.
Let's look at the first 10 natural numbers raised to the 4th power.
1^4 = 1 * 1 * 1 * 1 = 1
2^4 = 2 * 2 * 2 * 2 = 16
3^4 = 3 * 3 * 3 * 3 = 81
4^4 = 4 * 4 * 4 * 4 = 256
5^4 = 5 * 5 * 5 * 5 = 625
6^4 = 6 * 6 * 6 * 6 = 1296
7^4 = 7 * 7 * 7 * 7 = 2401
8^4 = 8 * 8 * 8 * 8 = 4096
9^4 = 9 * 9 * 9 * 9 = 6561
10^4 = 10 * 10 * 10 * 10 = 10000
Notice that of the first 10 natural numbers raised to the 4th power, only 5^4 and 10^4 do not have a 1 or a 6 as its units digit. All other 8 numbers have a 1 or a 6 as their units digit.
Now think of 11. If you raise 11 to the 4th power, it will end in 1 just 1^4 does. 12^4 will end in 6 just like 2^4 does. This holds true for the 4th powers of all of the natural numbers. Every natural number that ends in 1 has a 4th power that ends in 1. Every natural number that ends in 2 has a 4th power that ends in 6, etc. Since 8 out of 10 4th powers have a 1 or a 6 in the units digit, then out of 1000, you get 8/10 * 1000 = 800.