How?

Firstly, the probability of getting the number 1 on a single die is 1/6 (since there are six faces and only a face has 1 marked on it, assuming of course, the die is fair too —> meaning any ‘numbered’ face has an equal chance to appear in any throw).

So, the probability of getting the number 1 on all five dice,
P(1) = (1/6)^5

Likewise, P(2) = P(3) = P(4) = P(5) = P(6) = (1/6)^5

Hence,

P(same number, 5 dice rolled)
= P(1) or P(2) or P(3) or P(4) or P(5) or P(6)
= P(1) + P(2) + P(3) + P(4) + P(5) + P(6)
= (1/6)^5 x 6
= (1/6)^4
= 1/(6 x 6 x 6 x 6)
= 1/1296

as required, QED.