To solve this problem, we start with the given probabilities:

- The probability that the animal lives to be $20$ years old is $0.8$.
- The probability that it lives to be $25$ years old is $0.5$.

We are asked to find the probability that an animal, which is currently $20$ years old, will live to be $25$ years old. This can be found by dividing the probability of living to $25$ by the probability of living to $20$, because we are conditioning on the event that the animal has already lived to $20$.

Therefore, the calculation is as follows:

$$\text{Probability} = \frac{\text{Probability of living to 25 years old}}{\text{Probability of living to 20 years old}} = \frac{0.5}{0.8} = \frac{5}{8} = 0.625.$$

Thus, the probability that the animal, currently $20$ years old, will live to be $25$ years old is $0.625$.

So, the correct answer is $\boxed{B}$.