You don't really need to understand it. It's a phonology exercise. It's just a series of unnconnected utterances, designed to test how well you can pick up on linking. Linking is when the sound at the end of one word gets attached to the beginning of the next one (usually a word beginning with a vowel).
For example, the first three sentences:
1. I'm staying with an old aunt of mine in Leeds.
As you can hear, the five words in the middle become linked, so that it sounds like 'I'm staying withunoldauntuv mine'.
Listen to the next two sentences and notice where links happen:
2. My old uncle's coming to stay with us for Easter.
3. I often eat an egg with a handful of olives in the evening.