I haven’t watched the video of Bhante’s talk, but in the context of the Noble Eightfold Path, nekhamma comes under the second factor, Samma Sankappo, right intention, which also has non-ill will and non-cruelty as right intentions . It refers specifically to renunciation of sensory experiences, i.e. things that stimulate the senses. Being able to let go of sense objects is what allows us to get onto the meditation cushion in the first place, and once we sit down, it helps us stay there! When we practice this type of renunciation, we are turning away from the sensory world to an inner world, we move away from the grosser pleasures of the senses to the more refined pleasures of meditation.
Perhaps my favourite sutta about the importance of renunciation and right intention is the Two Types of athought sutta, MN19 where the three wrong intentions of sensuality, ill will and cruelty are introduced firstly:
Then, as I meditated—diligent, keen, and resolute—a sensual thought arose. I understood: ‘This sensual thought has arisen in me. It leads to hurting myself, hurting others, and hurting both. It blocks wisdom, it’s on the side of anguish, and it doesn’t lead to extinguishment.’ When I reflected that it leads to hurting myself, it went away. When I reflected that it leads to hurting others, it went away. When I reflected that it leads to hurting both, it went away. When I reflected that it blocks wisdom, it’s on the side of anguish, and it doesn’t lead to extinguishment, it went away. So I gave up, got rid of, and eliminated any sensual thoughts that arose.
And then the right intentions are introduced
Then, as I meditated—diligent, keen, and resolute—a thought of renunciation arose. I understood: ‘This thought of renunciation has arisen in me. It doesn’t lead to hurting myself, hurting others, or hurting both. It nourishes wisdom, it’s on the side of freedom from anguish, and it leads to extinguishment.’ If I were to keep on thinking and considering this all night … all day … all night and day, I see no danger that would come from that. Still, thinking and considering for too long would tire my body. And when the body is tired, the mind is stressed. And when the mind is stressed, it’s far from immersion. So I stilled, settled, unified, and immersed my mind internally. Why is that? So that my mind would not be stressed.
The Buddha uses a simile of a cowherd needing to keep watch of his cows in the case of the wrong intentions and being able to relax and rest in the case of the right intentions… it’s a nice image to remember.
Here’s the full sutta:
https://suttacentral.net/mn19/en/sujato?lang=en&layout=plain&reference=none¬es=none&highlight=false&script=latin
You can also read bhikkhu Bodhi’s chapter on Samma Sankappo in his excellent guide to the Noble Eightfold Path below.
I think the confusion here is that in english we also use the same word, renunciation, when referring to ordination, but that is not the same meaning here. So, yes, lay people can practice renunciation. It’s what folks do when they go on retreat, or devote time to meditation practice etc. And no, lay people don’t need to renounce all worldly affairs. There are plenty of examples of lay people practicing the path whilst having jobs, being wealthy, being queens and kings and parents and so forth, and they eventually became enlightened.