Why do we need a smaller sample size to detect a smaller proportion? The plot below shows the sample size needed to detect a proportion with a precision 0.01 for various true proportions:

This assumes an infinite population size, and the confidence intervals are fixed at 0.95.
If the true proportion is very small (or equivalently, very large) then we need a sample size of under 1000 to detect the proportion. However, if the true proportion is 0.5 then we need a sample size of almost 10,000.
I understand mathematically why this is from using the formula to calculate population size. But I would like an intuitive understanding of why this is the case.
It seems more intuitive to me that small proportions would need larger samples to detect them. Is this because I have kept the precision fixed (1% ± 1% is a much larger percentage error than 50% ± 1%)?

The critical part is "assuming the true proportion is 0.5", had the hypothesis been that the true proportion is 0.1 then the image would look different and centered around 0.1. The logic is relatively simple.
If you assume that the true difference is 0.5 and you get a result from your sample far from this number then you can be relatively confident that the true proportion is in fact not equal to 0.5. However if the result from your sample is close 0.5 then you are less confident, because of the inherent variability associated with this estimate (standard error). Hence, in this case, you need a bigger sample to decrease the standard error and be equally confident that the difference is not equal to 0.5.
To give a concrete example, assume a coin that you flip 10 times. If the coin is fair then the likelihood of you getting 9 tails in 10 tosses is very low, so you would probably conclude that the coin is in fact not fair. However if you obtain 6 tails from 10 tosses you cannot be as sure in this statement. You would need to throw the coin many more times to see if it is slightly biased.
As you put it, because you keep the precision fixed, the sample size will vary depending on how "close" you get to the hypothesized difference.
