It seems like such a huge risk to take given how expensive war is. It was much more risky for the banks making the loans since they had no way to enforce the payments. All Edward III had to lose was his reputation and it might make it more difficult to secure loans in the future.
So why did the banks agree to make these loans?
Just finished a research paper on Italian bankers in England in the late 13th century under Henry III and Edward I so I think I can provide some info. In general, the allure for many of the Italian merchant firms that gained a presence in English finances arose from all the possibilities for income within trade and the privileges that came with the attachment with the English monarchs.
It's first important to note that there was an established precedent for the presence of Italian bankers to be in close relationship to the monarch at the time. This trend began with Henry III's expanded use of the merchants and Edward I's (His successor) systematized use of merchant firms beginning with the Riccardi of Lucca in the late 13th century. For both cases, the bankers produced excellent results for the monarch, save for when the Riccardi lost their position due to being unable to finance a series of Edward I's wars in 1294.
Going off this idea, though I'm not acquainted with the reign of Edward III too well, its reasonable to suspect that the previous amounts of success with this policy should provide a reasonable cause for why the Italian bankers decided to finance the wars of Edward III.
Customs in particular was the crown's main support for repaying the loans. Essentially, the merchant firm attached to the crown at the time (Riccardi, Frescobaldi, and probably whatever one was working with Edward III at his time) would be given near complete control over the collection of customs fines and trade taxes. Considering the value of the English wool trade, collection would've been extremely lucrative for the Italians.
In conjunction, the bankers' position relative to the monarch would've added further privileges beyond customs. This often included exemptions of fines whenever customs or other regulations were violated. Most important though was that these firms would be relied on first-most for loans to finance whatever, which included war, meaning they would have a steady demand to supply funds.
If you want any further reading on the subject I'd recommend R. Kaeuper's Bankers to the Crown: The Riccardi of Lucca and Edward I which examines the first wide use of these banking groups under Edward I.