They received a mathematical assurance that it would work. The relying nodes, blockchain, miners and priority rules are all well-known and straightforward to account for. If you pay enough fees given your transaction priority, it'll get into a block eventually. If you want it to be mined sooner, you can pay higher fees.

And even if it won't be mined, the worst case scenario is that you would keep your Bitcoins and can try re-sending them later, they won't disappear to anywhere.