From what it sounds like, Free Companies are the player-run companies they talked about before. Right now we just have Grand Companies, where you pick a faction and complete content for it. In the future, players will be able to create their own companies, which also have their own content. So instead of having to join up with people from the Maelstrom or Twin Adder or the Immortal Flames to do something, you and your friends can start your own company and complete content that is designed for Free Companies.

The summoning thing Yoshida mentioned was in regard to the primals being more akin to gods than previous FF summons, so they would be more difficult to summon. In this case, a Free Company may possess the means to summon a primal, but it takes weeks of players charging their summoning gauges to be able to summon him together. So say there's a big boss your Free Company is planning on taking down, you'd pick a date when you're gauge is going to be full so you can unleash some Hellfire as a super attack to take the boss down easier.

That's what I got out of it at least.