They originate from Latin words relating to breathing in, breathing out, and breathing together.
The English word inspire comes from the Middle English word enspire, meaning "to fill (the mind, heart, etc., with grace, etc.)", which comes from Old French inspirer which comes from the Latin word inspirare meaning "to blow into, breath upon", figuratively "inspire, excite, inflame".
The English word expire comes from the Old French expirer, meaning to "elapse", which comes from the Latin exspirare, meaning to "breathe out, blow out, exhale; breathe one's last, die,", figuratively "expire, come to an end, cease".
The English word conspire comes from the Old French conspirer, which comes from the Latin conspirare, meaning literally "to breathe together", figuratively "to agree, unite, plot".