Understanding and proving the asymptotic behavior of $E[\min(\sum_i X_i, k)]$ for i.i.d. Bernoulli random variables

Question

I came across the following equation, where $X_i$ are i.i.d. Bernoulli random variables with probability $\dfrac{k}{n}$ of being $1$:

$$\mathbb{E}\left[\min\left(\sum_{i=1}^n X_i, k\right)\right] \approx \left(1− \frac{k^k} {e^k k!}\right)k, \mbox{ as } n → ∞$$

I would like to understand why this equation holds and how to prove it. Can someone please help me understand the intuition behind this relationship and guide me through the steps to prove it?

Any help is greatly appreciated. Thank you!