Over the years I have made many visits to different institutions, hosted many visitors, and met with many visitors at my home institution. These interactions have varied greatly in their value and success. Some have been incredibly interesting and fruitful. Indeed, many of my best new research ideas have had their beginnings in such discussions. On the other hand, some of the meetings seem to be rather "slow" and a waste of time. So here are a few thoughts on making the most of these meetings, from both sides.
The better prepared you are the greater the chance of a productive meeting. You want to find some common ground and common interest, i.e., something they have done you need to know about or something you have done you would like them to know about.
A minimum preparation is to scan the titles of the publications of the person you are meeting with. This will hopefully help find some common interests. Perhaps pick one paper that you would most like to ask them about.
Bring some "props" to the meeting. A printout of a recent talk you gave or a few powerpoint slides can help focus discussion. But don't rehash the whole talk. Share just a few highlights to gauge interest. If they want to know all the technical details they will ask. Bringing a copy of a recent paper to give away is good. They may even read it on the flight home!
Be a good listener. Be more eager to hear about their work than talk about your own.
For junior people meeting with senior people these meetings can be important career wise. Creating a good impression may lead to an invited talk at a conference, a sympathetic review of your next paper, or even a job offer. Giving a bad impression may lead to ... none of these. Hence, being well prepared is a good investment.
Overall I think everyone's goal should be to learn at least one interesting new piece of science.