Meeting is gathering, gathering, gathering. Leaving means leaving, leaving, being separated. People who meet will leave, which means there will be a party, people who meet will leave, and people who meet often will leave. It means that things change, and everything has an end. Meeting everyone is a once-in-a-lifetime fate.
These two sentences are not related in origin, but they have similar meanings, and people often use them together. According to Buddhist terminology, even people you see often will eventually leave. One prayer at a time is usually understood as meeting one at a time, that is, people in the world see each other less once. Generally speaking, the meaning of these two sentences is that things are impermanent and we should cherish the people we meet.
The first half of the sentence "Everyone you meet will leave" is a Zen saying. "Nirvana Sutra" says that "the prosperity of a husband is bound to decline, and there must be separation." And Buddhists say: people who often meet in the world will be separated, and those who meet will eventually be separated. Life is like a forest. Some people go out and they will never see each other again. The second half of the sentence "one prayer for one session" is a variant of "one session for one session" in Japanese Zen. Different people have different understandings of the phrase "one phase for a while". One stage means a lifetime, and the other stage means abandoning "self-attachment" and focusing on the present without distraction.