Caveat: I don't work at Strike. And this is pure speculation based on my current understanding of the situation.

Visa and MasterCard are networks. Strike is also a network (well, Lightning is the network but I'll come to that).

So what's a (business) network? Imagine MasterCard deciding to onboard a new country. They have to go to every shop and store and have them sign up to be a part of their new network. Perhaps there was already a local network going on so they either acquire it or enter a partnership with them. This is what a business network is. It's the rails that allow two unfamiliar parties to enter in a deal (i.e. financial settlement).

Imagine what it would look like if these networks did not exist. If you're from the UK and wanted to travel to Madagascar, there was no way for you to use your credit card. The network just didn't exist there. You'd have to bring cash and exchange it for the local currency. This is what such a network allows. The merchant trusts MasterCard to pay them for the product/services, and in turn MasterCard trusts you to pay MasterCard back (at some point).

So what about Strike? Building a network is hard work. Takes a lot of time and money. There's a strong first-mover advantage, which is a characteristic of any network (LN anyone?) given that each node is as valuable as its connections (LN channels anyone?) and that for new nodes to come online they have to connect to existing nodes (d'uh), making the first nodes naturally more well-connected than nodes that come later.

So what about the LN allowed Strike to enter this very competitive market? When Strike went to the companies that own the Point-of-Sale machines (the "credit card" machines at the tilt), they weren't just bringing along their own prospect customers. They were bringing along all the LN users with them. That's what allowed them to close the deal. You wouldn't enter into a deal that might anger the big Visa and MasterCard players just for fun. You'd need to have strong reasons. They not only had the argument of having every LN user with them, but also the much stronger argument that the LN is an OPEN PROTOCOL that is growing exponentially, which means there will be more and more integrations and more and more users/customer in the foreseeable future, as the protocol is open for anyone to join.

Hope this helps to understand a little bit better what exactly is it that Strike is building and how. Again, this is mostly personal speculation based on videos of Jack Maller's talks.

Now, how on earth is Strike able to buy/sell Bitcoin without fees? That, I don't know. I've heard something about being able to trade liquidity for optionality (kinda like how you can measure an atoms position or speed but not both I guess) but I can't say I understand it.