What does "$n$-fold eigenvalue" mean? I am trying to find the eigenvalues of the matrix $A = I + N$ where $N^2 = 0$. The answer I have says

$1$ is the $n$-fold eigenvalue.

I am not sure what is meant by "$n$-fold" but my guess is that $1$ is the only eigenvalue and is repeated $n$ times. Indeed, when I have the $2$ by $2$ matrices
$$I = \begin{bmatrix}1&0\\0&1\end{bmatrix}$$ and $$N = \begin{bmatrix}0&1\\0&0\end{bmatrix}$$
the only eigenvalue of $A$ is $1$ and it is repeated $2$ times. 
Is there a way to prove this for $n$ by $n$ matrix?

You have interpreted "$n$-fold" correctly.
We have $(A - I)^2 = 0$.  However, if $\lambda$ is an eigenvalue of $A$ and $x$ is an associated eigenvector, then we must have
$$
(A - I)x = Ax - x = \lambda x - x = (\lambda - 1)x,
$$
and similarly
$$
(A - I)^2x = (A - I)[(A - I)x] = (A - I)[(\lambda - 1)x] = 
(\lambda-1)[(A - I)x] = (\lambda - 1)^2 x.
$$
It follows that 
$$
(\lambda - 1)^2 x = (A - I)^2 x = 0 \cdot x = 0.
$$
