Universal hashing function probability
Let’s begin by talking about the intuition for universal hash families. Intuitively, a family of hash functions is universal if for any distinct objects x and y that you’d like to hash, if you select a random hash function from the hash family, the probability that you get a collision between those two elements is at most 1/m, where m is the number of buckets. In other words, universal hash families tend to spread elements out in a way where the probability of a pair colliding is the same as if the elements were distributed randomly.

Let’s see how the definition accomplishes this. Here’s the definition from your question:


  H is universal if &forall; x, y &in; U where x &ne; y | { h in H : h(x) = h(y) } | = |H| / m.


For starters, I’m assuming that we’re talking about hash functions that map from some set U to the integers 1 through m or 0 through m-1. With that in mind, let’s unpack some of this notation.

If we replace the universal quantifier (&forall;) with the plain English “for all,” this definition says “H is universal if for every choice of two different items x and y to hash, some inequality is true.” So let’s look at that inequality. First, what is this bit?


  | { h in H : h(x) = h(y) } |


The vertical bars here represent the size of a set, and the set in question is this one:


  { h in H : h(x) = h(y) }


Read literally, this is the set of all hash functions h in the family H where h(x) = h(y). Keeping in mind that we’re talking about hash collisions here, we can think of h(x) = h(y) as saying that hash function h causes x and y to collide with one another (have the same hash code). With that in mind, the complex expression 


  | { h in H : h(x) = h(y) } |


means “the number of hash functions in H where x and y collide.”
Combining that with our earlier bit, we can rewrite the entire definition as


  H is universal if for any two distinct elements x and y that we want to hash, the number of hash functions in H where x and y collide is at most |H| / m.


So what’s |H| / m? That’s the total number of hash functions (|H|) divided by the number of possible outputs for any one single hash function (m). It might help to divide the entire expression through by |H|, which would then give this final definition:


  H is universal if for any two distinct elements x and y that we want to hash, the probability of picking a hash function h where x and y collide (that is, the total number of hash functions where x and y collide divided by the total number of hash functions) is at most 1/m.


Hope this helps!