It refers to the fact that, sometimes, the absence of a person or a thing is more noticeable than their presence. It is used also in an ironic way

I finally agreed on a big birthday party for my twins, but when it came to cleaning up after the party, both of them were conspicuous by their absence

Alla fine ho acconsentito a fare una grande festa di compleanno per i miei gemelli, ma quando si è trattato di ripulire la casa dopo la festa, tutti e due brillavano per la loro assenza

At the party for Renato's retirement, one person was conspicuous by his absence: Renato, the guest of honor. His coworker, who planned the event, forgot to invite him!

Alla festa per il pensionamento di Renato, una persona brillava per la sua assenza: Renato, il festeggiato. Il collega organizzatore si era dimenticato di invitarlo!

One name conspicuous by its absence in the list of the persons to thank for winning the championship, was that of the resigning coach, being at odds with the owner

Nell'elenco delle persone da ringraziare per la vittoria del campionato, brillava per la sua assenza il nome dell'allenatore dimissionario, ormai ai ferri corti con la proprietà

The idea goes back to Roman times. The historian Tacitus (Annales III, 76) tells that at Junia's funeral procession, the portraits of Cassius (her dead husband) and Brutus (her dead brother) were missing, which made them even more visible: