Lets consider a graph where each node represent a person and an edge denotes that two person have done handshake. So, our problem reduces to finding the number of nodes in a graph with odd degree and to see if this is even.
Each edge in a graph contribute 2 to the sum of degrees. i.e.,
Sum of degrees in a graph $ = 2e.$
For any vertex in the graph, its degree is either odd or even. So,
Sum of degrees in a graph = Sum of degrees of odd degree vertices + Sum of degrees of even degree vertices
Sum of degrees of odd degree vertices = Sum of degrees in a graph - Sum of degrees of even degree vertices
Here, both terms on RHS are multiple of 2, as first one is $2e$ and sum of even numbers is always even. The difference of two even numbers is also even, and hence sum of degrees of odd degree vertices in a graph is even. We get an even number when we add odd numbers "even" number of times and not otherwise. So, number of vertices with odd degree in any graph must be even.