As you stated, transitivity is that overall $x \succeq y$ and $y \succeq z$ implies $x \succeq z$. I will show an example where majority rule isn't transitive and hopefully it will answer your question.
Imagine that we live in a world with three people: Person 1, Person 2, and Person 3. Each of these people have preferences over three outcomes $x$, $y$, and $z$. Every decision in this world is made according to majority rules, i.e. The outcome chosen is the one preferred by at least two of the individuals.
Now imagine that the three people have preferences according to:
- Person 1: $x \succeq y \succeq z$
- Person 2: $y \succeq z \succeq x$
- Person 3: $z \succeq x \succeq y$
The majority rule decision making implies that we rank the pairs according to the following:
- $ x \succeq y$ because Person 1 and Person 3 prefer $x$ to $y$
- $ y \succeq z$ because Person 2 and Person 3 prefer $y$ to $z$
- $ z \succeq x$ because Person 2 and Person 3 prefer $z$ to $x$
Notice that this is exactly a violation of transitivity.