If it is very natural for a person to talk about themselves and their business, how can you be patient and slip a salient point in to solve someone else's problems?

As noted, patience is a virtue. Without patience we would move at break-neck speed without taking time to reflect on others' thoughts and words. In order to really understand what they have said, we have to firstly spend time listening, and then we must practice patience before talking ourselves, giving us time to assimilate what was said. And listening patiently means waiting until they have finished talking. It can sometimes be very annoying when a person goes on and on. This person is not likely aware that they are boring those around them, but if you take time to really listen to what they have to say, you may learn something new. The person is not likely aware that they are taking all the air time in the room either. In this situation, it is best to allow them to talk while interjecting a couple of questions now and then.

It may seem like you are egging them on (and in a way you are) but in reality you are practicing patience by allowing them to talk with you, steering them in the direction you would like them to go. Even as they are talking you should be making mental (or written notes) about their topic - especially if it has to do with some business pain that your company can solve for them. Being patient is probably one of the most difficult things to do, but it has its own rewards for the listener. Just think about a time when you waited in line and ended up being the last one served just before the teller closed and they had some left over goodies for you to take. The patience paid off.

So you have asked the questions, spent your time listening to the other person and secretly taken mental notes on the answers. Now it is time to make a decision as to whether there is a possibility of doing business with this person. If so, it is not a simple question of asking for a meeting, and definitely not simply collecting their business card. Everything is timing, and this is about timing the best way to exchange information.