The preposition 'out' is deliberately supplied in the English translation of κατεργάζεσθε to form the verb phrase 'work out' to show how the context defines it because Philippians 2:12 itself wasn't saying that we should work for our own salvation in the sense of saving ourselves from what Christ had saved us from.
and be found in him, not having a righteousness of my own that
comes from the law, but that which comes through faith in Christ, the
righteousness from God that depends on faith—
Philippians 3:9 (ESV)
12 Therefore, my beloved, as you have always obeyed, so now, not only
as in my presence but much more in my absence, work out your own
salvation with fear and trembling,
Rather, Philippians 2:12 was saying that we need to express or show the salvation we both personally and freely received by obedience.
12 Therefore, my beloved, as you have always obeyed, so now, not only
as in my presence but much more in my absence, work out your own
salvation with fear and trembling,
Philippians 2:12 (ESV)
For by grace you have been saved through faith. And this is not
your own doing; it is the gift of God, not a result of works,
so that no one may boast.
Ephesians 2:8-9 (ESV)
The verb phrase 'work out' (v.12) coheres with the verb phrase 'work in' (v. 13) showing that both God and man work together (synergistically) to accomplish one purpose. In the former, the believer works out or reveals his/her salvation by continued obedience while in the latter, God is working from within the believer, enabling him/her both to will (desire) and to work (out his/her salvation v. 12).This is the sense which James 2:18 explicitly conveys ("I will show you my faith by my works").
12 Therefore, my beloved, as you have always obeyed, so now, not only
as in my presence but much more in my absence, work out your own
salvation with fear and trembling,
13 for it is God who works in you, both to will and to work for
his good pleasure.
Philippians 2:12-13 (ESV)