Origins
Present is three words wearing the same spelling. A gift. A moment in time. The state of being here. All three descend from Latin praesēns — 'being at hand, being before one' — the present participle of praeesse: prae- ('before') + esse ('to be').
The unifying idea is proximity. What is present is before you — visible, available, now. The temporal present is the slice of time before you at this instant. A gift is a present because it is presented — placed before the recipient. A person who is present is before others, in their company.
The verb to present (stressed on the second syllable) preserves the act of placing before: to present evidence, present a case, present a bouquet. A presentation is a formal placing-before-others. Represent is to place before again — to stand in for something or someone.
Latin Roots
The mirror word is absent, from Latin absēns — ab- ('away from') + esse ('to be'). If present means 'being before', absent means 'being away from'. Presence and absence are the two poles of the verb 'to be' in Latin.
The PIE root *h₁es- ('to be') is the most fundamental verb in Indo-European. From it came the Latin esse, the English is, the German ist, the Sanskrit asti, and the Greek esti. Present packages this ancient root in a prefix that fixes it in space and time: not just being, but being here, being now.