If I send bitcoins to somebody in order to buy a product or a service, how can I demonstrate that I am the actual sender of the money?
You don't. All you need to demonstrate is that the money was sent for the purpose of paying for that particular order, which the merchant can do automatically by using a unique Bitcoin address for each order. If that address receives the requisite number of bitcoins, the order is paid for and the merchant ships it out, regardless of who actually paid.
The reason for doing it this way is that in many cases, you are
not the actual sender of the money, nor will the actual sender of the money necessarily know anything about the order that they're paying for. This will happen if you are using a web wallet or other payment processor that makes Bitcoin transactions on your behalf. In that case,
they're making the payment, not you, and you cannot prove otherwise.
Note that if a merchant
does ask you to prove you are the sender, they're doing something hopelessly wrong and you should not do business with them.
