The worse case is 5% because "is a bot" is a boolean variable. That has a standard deviation of 0.5 in the very worst case (50% on value 0, 50% on value 1, so the mean is 0.5, and the deviation can't be more than 0.5). And the standard deviation of a random sample scales like 1 over the square root of the sample size, so 0.5 divided by 10 => 0.05 (5%).

For presidential elections it's common to see samples sizes like 1,000 or so, which have standard deviations of around 1.5%. That's better than 5%, and it makes sense since elections are often won by just a few %. But here, IIUC, the goal was to see if bots are a big part of Twitter or not, and so the answer "there are fewer than 5% bots" is enough. That is, we don't care if the % of bots is 3.5% or 4.7%.