to pour out as a drink offering: to be poured out like wine spilled before God as an offering
In ordinary Greek life this was the most common act of worship there was: before drinking, a person poured a little wine from the cup onto the ground or an altar as a small gift to a god. The Old Testament had the same gesture. Alongside the main animal offering, an Israelite poured out wine at the base of the altar, and the Greek translation of the Old Testament used this very word for it. So the word carried a worship meaning that any reader, Jewish or not, would recognize: a measure of wine poured out and given up.
Paul uses the word only twice, and in both places he turns it on himself. He is not the priest doing the pouring; he is the wine being poured. In Philippians he pictures the Christians' faith as the real offering and his own life as the drink offering poured out on top of it, the small extra portion that goes with the larger gift. In Second Timothy he says the pouring is already happening and his death is near. The shade of meaning is the same in both: a life given up and poured out in service to God and to others.
Two things give the picture its weight. First, the form Paul uses means he is being poured out, not that he is pouring. Someone else holds the cup, and that someone is God. Paul's part is simply to be willingly emptied. Second, this is a picture of devotion, not of paying for sin. Scholars are agreed that Paul is not calling his death a sacrifice that takes away sin (he reserves that for the death of Christ); he is describing a life freely spent for the gospel and for the churches he served.