Confidence Limits are the upper and lower values in a Confidence Interval. A Confidence Interval is a range of values that is expected to contain the true value of what is being measured.
Let's say you were reporting the average weight of a one-year old baby. Let's assume that the true value was 10kg, meaning if you were to weigh every single baby in the country, that would be the value that you would get. Since it would impractical to do this, you decide to get a sample of 100 babies, giving you an average weight of 9.5kg. This is slightly different from the true value, but close enough.
Why the discrepancy? Because no sample can perfectly represent the entire population. In fact, if you were to get 3 samples of 100 babies each, you could expect to get 3 different average weights. This is because every sample is slightly different from other samples, even if they all belong to the same population. For this reason, it is useful report results in terms of Confidence Intervals. A narrow Confidence Interval with Confidence Limits close to each other (for example, 9.5kg to 10kg) indicates that you are more precise, while a wide interval with Confidence Limits farther apart (like 7kg to 11kg) indicates less precision. In both cases, the true value of 10kg still falls within the interval.