No, you cannot use this app to buy bitcoins. So I'm afraid it does not solve one of the hardest challenges Bitcoin still faces, which is how to get any. You'll have to figure out the best/cheapest way depending on where you are.

If you happen, for example, to stand in front of a Bitcoin ATM, you can use that to turn fiat that you have in the form of cash into fiat that you have in your Bridgewalker account (by buying Bitcoin from the ATM and sending it to your Bridgewalker address), which you can then later spend wherever it says "Bitcoin accepted here".

A lot of exchanging back and forth for sure, but basically the idea of "Bitcoin only as tranfer mechanism" applied at every step. It's also just a technical prototype. In practice it might make sense to cut some of those exchange fees by having a more direct way to fund your account, at the cost of only being available in certain markets/countries. For this prototype I went with the easiest solution: only Bitcoin in, and Bitcoin out.

Presumably you can create a wallet and use this to send bitcoins to it?

Doesn't look like it. You fund it with bitcoins which it converts to euros