When you "exchange" something with a person, you give them something and they give something back to you. In the example above, the speaker suggests exchanging business cards. That means that each person gives the other person their business card.
Other words that people often use with "exchange" are:
- exchange phone numbers
- exchange ideas
- exchange a glance (this means to look at each other secretly to communicate a message)
- exchange words (this means to get into an argument)