This means some tiny change in x. So you're effectively computing the probability at an exact point since δx goes to 0. It's really just saying that p(x) is a function that computes the probabilities of variable x being any of its possible values.
The interval (a,b) part means you can compute the probability of x landing in an range from a to b by taking the integral of p(x).
Another way to phrase the whole paragraph is: If p(x) tells you the probability of x landing at a particular value, to compute the likelihood x lands in interval (a,b) just take the integral of the probability density function over that range.
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