When you vaccinate, your goal is to produce an adaptive immune response, a specific response to a specific antigen that will later be detected by the immune system if a real/fully potent infectious agent arrives.
Toll-like receptors, on the other hand, are part of the innate immune system that detects general characteristics of infectious agents, such as RNA or bacterial cell walls.
The innate immune system and adaptive immune system don't act completely independently, though. Activation of the innate immune system can stimulate the adaptive immune system. The "message" is: "hey, we've got an invader here!" Perhaps you can think of it a bit like a group of soldiers reporting to their command that they are under fire: they might not know who exactly is attacking them, or from where, and they're doing the best they can for the moment but looking for some fire support.
So, one problem if you vaccinate with just a specific part of a virus is, well, it's not really doing any damage. There's no inflammatory signal from dying cells that tells the immune system to be a bit suspicious; there's no telltale sign of viral infection from RNA floating around or bacterial infection from pieces of cell wall. There's an antigen present that the innate immune system could recognize and form a response to, but if no damage is taking place and no markers of invasion are taking place, it may be more prudent to stand down and wait it out.
That's the role of an adjuvant: it need not have any relationship to the intended target virus, it just has to trigger the message that something is going on and get the adaptive immune system on the case, looking for something out of place. When that happens, the other part of a vaccine, the part derived from part of an actual virus, gets noticed.