The following sentences are ambiguous. Provide two explanations which illustrate the different meanings of each sentence. We need more empathetic leaders. They are visiting relatives.

'Ambiguity' means a lack of clarity or concrete detail, which can make it difficult to discern meaning. In the case of these two sentences, the lack of context and detail make it possible to interpret them differently.
The first sentence, "We need more empathetic leaders," can be interpreted in a couple different ways. First, though, let's find the word that makes this sentence ambiguous by asking ourselves what needs more clarification. In this case, I would say the word most lacking in clarity is 'more,' which could mean we need more leaders or leaders who are more empathetic.
An average person might say this about government leaders they don't find empathetic, for example, to mean our leaders need to exhibit more empathy. Alternatively, the same person could say this to mean we have some empathetic leaders, but we need more of them. Without hearing the stress a speaker might place on different words aloud or having additional context, it's hard to discern exactly what this sentence is trying to say.
The second sentence, "They are visiting relatives," has a similar problem. Again, I would start by identifying the words that are most ambiguous—in this case, 'visiting.' To understand the sentence, we need to know if 'visiting' is functioning as a part of the 'are visiting' verb phrase or whether it's being used as an adjective to describe 'relatives.'
With 'visiting' used in this way, there are two possible meanings for the sentence. 'They,' the subject of the sentence, may be visiting relatives, with visiting, in this case, being part of the verb phrase. 'They' can also refer to 'visiting relatives,' meaning the sentence subjects are relatives who are visiting, with 'visiting' functioning as an adjective. Again, we need context or emphasis from a speaker to understand what exactly is being said here, which is what makes the sentence ambiguous.