The main cause of post-vaccination pain is the needle has physically damaged your body. You do have a new unplanned hole in your skin, muscle, fat layer, etc. It's very operator dependent.

Erythema is redness in the skin due to increased blood flow to the capillaries in your skin. Same as any skin injury.

Induration is your skin thickening/hardening after an injury. It is a result of inflammation response from a skin injury. Your body has only a handful of responses to an injury, and inflammation is an easy one.

Sub-cutaneous nodules, or little bumpy lumps under your skin. Usually from local inflammatory reactions or immune-mediated responses.

That is all dependent on how much volume was the dose, where it had to go, how quickly it diffuses away from the injection site.

Next is the adjuvant chemicals that are triggering the immune response.

There are different stimulants in different vaccines that force your body to recognize something has happened. It can induction of cytokines and chemokines (inflammation, hey, look over here), recruitment of immune cells (swelling, lets bring lots of workers to the site), enhancement of antigen uptake and presentation (swelling as it opens up your internal paths), and promoting antigen transport to drain stuff away into your lymph nodes (itching, got to physical push stuff into the lymph system).