What is a Loeb measurement or Loeb space?

I've been looking for a simple explanation, but all I find is stuff like this:

What is a Loeb space? In applications one usually only needs

to work with hyperfinite Loeb probability spaces in a nonstandard universe

which is a countably indexed ultrapower. In that case, a hyperfinite set has the

external cardinality of the continuum (the cardinality of the continuum will be

denoted by c), so there is a bijection between the hyperfinite set and the unit

interval [0, 1]. Using this bijection, one can impose the Loeb measure structure

on [0, 1]. Thus, a Loeb measure can simply be viewed as a measure on the unit

interval. In this setting, the special measure-theoretic properties of Loeb spaces

can be restated as properties of some measure on the most familiar underlying

space [0, 1].

I got into this by trying to understand Edward Nelson's "Internal set theory" (IST). I previously asked a question about that, but so far no responses.