Here's a little through experiment I use to come to this conclusion:
Let's say you start a counter from the number 0, and keep on incrementing it. The moment you stop it to look at the counter, is it going to be odd or even?
At any given moment in time, either the number of observed odd numbers is the same as the number of even numbers, or the number of even numbers is larger by 1 (such as going from 0 to 1 to 2). So in the end there's always a slightly larger chance at stopping on an even number.
I know it's more complicated, I use it just as an intuitive explanation.