OK
All guitar amplifiers have to have a load. If you think of your amp as being like an engine, your engine is designed to operate with a load. The motor needs to have some force working against it, otherwise it might run faster than it is designed to and would damage it.
Well, guitar amps are the same way. Your guitar amp is designed to have a certian resistance pushing back against it. This is the load. Guitar amps have 3 typical load settings, 16 ohms, 8 ohms and 4 ohms. A few amps out there can also run at 2 ohms.
You NEVER want to run your amp without a load. This can cause unrepairable damage to your amp.
So, what do you do when you want to run your amp without speakers? Well, you use a dummy load. A dummy load simulates the resistance of a speaker load. Some dummy load (like the THD hot plate) use a big resistor to simulate the load. The Weber dummy loads have a speaker inside them that doesn't have a cone.
I just threw the dummy load suggestion out there. I think it would probably be easier (and maybe cheaper) to try to run a headphone amp from the amp's direct out. But the only way I can think to do that would be to run the amp through a small mixer that has a headphone output.