How URL redirect can be done without changing URL?

I am asking to verify what I hear from my provider.

I just bought domain name (let's say foobar.com) and I would like to redirect it to helloworld.blog.com in such way, that if somebody enters foobar.com in browser, the content from helloworld.blog.com is shown but the address is not changed (foobar.com should remain).

When user would like to go to sub-pages address will change, because entire site (WordPress) uses hardcoded absolute URLs. Pity, but that's OK.

Previously I bought a domain at different provider, and I simply made 2 clicks to achieve this. Now I have different provider and he explains that even if such option is done by just clicking in reality there is small page created at original address (foobar.com) with frame pointing out to target site (helloworld.blog.com).

Is this true? I mean does redirecting URL (with URL kept) is done this way? I don't want to be accussed of traffic abuse or something like this (and besides it smells bad for me).

Or panel control at my current provider is too weak and missing crucial features?

Note: I don't have problems in technical terms of writing such proxy html page with frame with it, I am simply looking for a way to ensure/deny it is done this way for real.