Specifically, the cause of the crummy feeling is due to your innate immune system cells, like macrophages, detecting foreign materials and releasing cytokines to call attention from more and more immune cells (this starts the inflammation response). Some types of cytokines are specifically responsible for triggering the brain to make you sleepy, sore, and have a fever. Here's a great little video with a little more detail: https://youtu.be/gVdY9KXF_Sg
The fatigue is likely caused by fever. In response to an infection with a cold-causing virus, your immune cells begin to secrete cytokines that cause both localized inflammation and a full-fledged immune response from the rest of the body. Chemokines like Interleukin-1 (IL-1) and IL-6 are considered major endogenous (meaning they are produced by your own body) pyrogens (molecules that cause fever).
In addition, other endogenous molecules are also produced that aid in inducing a fever. IL-1 and IL-6 can both activate the [arachidonic acid pathway](http://www.cvphysiology.com/Blood%20Flow/BF013
), which ends in the synthesis of Prostaglandin E2 (PGE2). PGE2 is a major regulator of the febrile response and acts on the hypothalamus and other areas of the brain, which induces shivering to create heat, constriction of blood vessels to prevent heat loss, and can also cause the release of adrenaline.
The virus itself can cause tissue damage, which further exacerbates inflammation in the area of the infection. Not all viruses that cause colds replicate through the same mechanism: some are lytic and others are lysogenic. Lytic viruses like rhinoviruses rupture cells once the infected cell reaches its burst size. The insides of the cell can trigger inflammatory signals from neighboring cells as well, further exacerbating the immune response. Lysogenic viruses such as coronaviruses tend to bud or undergo exocytosis from cells, and the immune response to these is somewhat dampened by the lack of damage-associated molecular patterns (DAMPs) that would trigger inflammation from nearby cells.
Unfortunately, from what I can find it's not totally clear what it is about fever or inflammation that causes lethargy, though I would speculate it likely has to do with the modulation of the [autonomic nervous system](https://en.wikipedia.org/wiki/Autonomic_nervous_system
) by prostaglandins.