"Seize the day" is a translation into English of an old Latin saying, "Carpe diem." In the 1800s, when educated people knew Latin, it was common to use various Latin phrases as English idioms. Some examples include "et cetera" and "quid pro quo."
It is easier to understand if we change the wording slightly. Instead of saying "Seize," say "don't let it go." It means "don't let the day go by," "don't waste the day," "grab the opportunity."
Go get 'em! Those who snooze, lose. <---Phrases with somewhat similar meanings.