I’m going to include $29$ February as a possibility. If each of the $366$ possible birthdays is represented only $27$ times or fewer, we could have at most $27\cdot366=9882$ people. We actually have $10,000$ people, more than $9882$, so some birthday must be represented more than $27$ times. Thus, there must be at least one birthday shared by at least $28$ people.
Can we do better? No, because $28\cdot366=10,248$; we could, for instance, have $118$ birthdays each shared by $28$ people and $248$ each shared by $27$ people, since $118\cdot28+248\cdot27=10,000$. Thus, we can’t guarantee that there is a birthday shared by $29$ people.
This reasoning, when carried out in general, leads to the conclusion that if you divide $n$ things into $k$ categories, at least one category must contain at least $\left\lceil\frac{n}k\right\rceil$, where $\left\lceil\frac{n}k\right\rceil$ is the smallest integer greater than or equal to $\frac{n}k$.