Checkins are a a way of finding out where everyone is at, at the beginning of any meeting or gathering. Our emotional state affects our decision making and how we react to circumstances. If we check in we share where we are emotionally.
For example, if I have just had an argument over a totally unrelated issue, I'll probably be bringing some of that feeling into the room, maybe some anger and rejection. By 'checking this in' at the beginning of a meeting I not only give others a clue about what I might be sensitive about and external factors that might be influencing me, but also it helps me acknowledge my own feelings consciously as opposed to having them influence me just out of my awareness.
Checking out is also a good way of reflecting the impact of the meeting and noticing any change or what you might be taking away with you into whatever you are doing next.
An example of a check-in using the above context might be: 'I'm feeling a bit angry and maybe a little sensitive and a bit sad because this morning as I was leaving home I had an argument with my wife about something that feels trivial and now I'm worried about what will be waiting for me when I get home tonight.'