IQ to percentile
A percentile states the rank directly: the percentage of the population scoring at or below a given IQ. These conversions come from the normal distribution that the 100/15 deviation scale is defined on.
The table
Read it as: someone scoring this IQ scores at or above this share of people. The right column restates the same thing as a count out of a thousand, which is often easier to feel.
How to read a gap
Equal gaps in IQ are not equal gaps in rank. Near the middle of the scale a few points move you past a lot of people, because that is where everyone is bunched. Out at the tails, several points barely change the count, because there is almost nobody left to pass.
From 100 to 110 you pass roughly a quarter of the population. From 130 to 140 you pass around two people in a hundred. The second jump is the larger one in ability terms and the smaller one in rank terms.
Precision at the tails
The percentiles above are exact arithmetic on the normal curve. They are not a statement about how precisely any test can place you. A fixed-length battery has few items hard enough to separate people above about 140, so the measurement error there is wide even though the table looks confident.
This is the practical reason a result should carry an interval. A reported 138 usually means something closer to “somewhere in the low-to-high 130s”, and the scoring method states that explicitly rather than hiding it behind a point estimate.
For what the bands themselves mean, see what IQ scores mean.