I will try to expain here in a simple way. 1/0 is read as one divided by zero and division is always defined in terms of multiplication. Now keep in mind division and multiplication properties and read following...

Take
a
--- = x
b
OR b × x = a

Let b is 0 and a is 1.

0 × x = 1

What will be the value of x if b is 0 and a is 1.

i.e.
x = ? when 0 × x = 1

There is no value of x such that 0 × x = 1, because 0 × x always result 0 for all x. In other word x times zero always result 0 or anything times zero always result o and never be equal to 1.

x = not exist when 0 × x = 1

Its mean 1/0 is not exist, or is not defined, or is undefined. Similarly any non-zero number divided by 0 is undefined.