It doesn’t say there’s a number k: it says that there is a number $k$ that has a certain property. You can’t just pick any old number $k$: it may not have the required property.
Suppose that you have an algorithm that always takes $2^n$ steps to reach its conclusion when the input is of size $n$. Then no such $k$ would exist: no matter what constant $k$ you choose, $2^n>n^k$ for all sufficiently large $n$. You cannot find a constant $k$ such that $2^n\le n^k$ for all $n$. Therefore this is not a polynomial-time algorithm.
Now suppose that you have another algorithm that takes $2n^2+3n+4$ steps to finish when the input is of size $n$. For all $n\ge 4$ we have $2n^2+3n+4<n^3$, so for this algorithm we can choose $k=3$. (The definition is a little inaccurate: it should say that the algorithm reaches its conclusion in fewer than $n^k$ steps for all $n$ greater than some minimum size.)