Definition
The sampling distribution of a statistic is the probability distribution of the values that statistic would take if you computed it on every possible sample of a given size from the population. For example, the sampling distribution of the sample mean from a population with mean mu and standard deviation sigma is (approximately) normal with mean mu and standard deviation sigma / square root of n.
It is the bridge between probability and statistics: data become a random variable through the lens of sampling, and that random variable has a distribution you can reason about.
Why it matters
How it works
In practice you never see the full sampling distribution — you only see one sample. But probability theory tells you what the distribution looks like under the assumed model. For a sample mean, you know its mean is the population mean and its standard deviation is sigma / square root of n. The CLT tells you the shape is normal for large enough n.
This theoretical knowledge is what lets you reason backward from the single sample to the population: 'Given the sampling distribution of the mean, my observed sample mean is consistent with population means in this range...' That is the essence of frequentist inference.