Why do those with viral infections feel better after wrongly taking antibiotics? Though I'm unversed in science, I accept and understand that bacteria differ from viruses; so antibiotics don't help in viral infections. Regardless, because older family members still allege that antibiotics have mitigated their colds, can psychology explain their misbelief? 

They claim that within a day of taking antibiotics, their ability to speak returns and  their sore throats are soothed. I can attest to this improvement; they do feel better. I'm also unversed in psychology, but can some placebo effect (again, NOT the antibiotics themselves) truly contrive such genuine relief?
Placebo effect is not the only explanation, although it can be a large component. 

To be clear: Neither of the explanations below are reasons to use antibiotics for infections not due to bacteria. They do illustrate anchoring. Just because we call a drug an antibiotic and first note its effects to be killing bacteria does not forbid that drug from having other effects. 

Antibiotics are anti-inflammatory

Bacterial and viral infections both induce an innate immune response, which includes the release of pro-inflammatory cytokines. Antibiotics can provide symptomatic relief by inhibiting the release of those cytokines, decreasing inflammation. 

Bacterial super infections

The risk of a bacterial infection jumps during and shorty after a viral infection. Children, for example, are more likely to get strep throat after the flu. If the patient is unaware of two infections, taking an antibiotic will appear to treat the viral infection. It is, in reality, treating the bacterial rider.