This quote suggests a philosophy of hospitality and openness, but also of temporality and letting go. It refers to the ethos of welcoming the new ("welcome the coming") and embracing change ("speed the going guest"). The speaker is saying that they value the guidance of Homer, the ancient Greek poet, who often emphasized the importance of hospitality in his works.

"Welcome the coming, speed the going guest" is a metaphorical way of saying that we should be open to new experiences, people, ideas, and opportunities, welcoming them with open arms. However, we should also be ready to let them go when the time comes, not clinging to the past or trying to make things stay the same. This is a philosophy of adaptability, flexibility, and acceptance of change.

In today’s world, this quote could be applied in many contexts. For instance, in personal development, it could mean that we should be open to learning and growth, always ready to welcome new knowledge and experiences. However, we should also be willing to let go of old habits, beliefs, or ways of thinking that no longer serve us. This could also be applied in business, where companies need to be innovative and open to new ideas, but also ready to let go of outdated practices or products.

In relationships, it could mean that we should be open to meeting new people and forming new connections, but also ready to let go of relationships that are no longer healthy or beneficial. In a broader societal context, it could mean that we should be open to societal progress and change, but also ready to leave behind outdated norms or traditions.

Overall, this quote is about the balance between welcoming the new and letting go of the old, and how this balance can lead to growth, progress, and personal development.