I think there are two parts to your question.
1. Why do both sentences use the verb 'to have' and the verb 'to be'?
We use 'have' as an auxiliary verb when making certain tenses and in these cases the verb 'have' has no meaning really. It only tells us the tense. In this case we use 'have' and the past participle to form the Present Perfect. We choose this tense when we want to communicate experience or knowledge (based on past experience) AND when it's not important when/how/why something happened. So for example "I have eaten lobster." - it is not important when, the important information is that I know what lobster tastes like.
2. The verb 'to be' is used in both sentences, but in slightly different ways. So we say "I am English." or "I am tall." to describe a quality of mine (not an ability - we normally use "I can..." for abilities). BUT we also use 'be' as a synonym for inhabit/visit/exist and we say "I am in Rome today."
Therefore the first sentence - "I've been this guy" - means "I was like this in the past, and so I understand what it is like to be this kind of person."
The second sentence - "I have been with someone" - literally means "I was in the same place as another person, and therefore I understand what it is like to be in the same place as another person."
We often use the phrase "to be with someone" in an idiomatic way. It often means "to be in a relationship with someone" - or in a more vulgar sense it can also mean "to have sex with someone" - you will need to refer back to the original TED Talk to see how it is being used in this context.
Hope this helps