What does "I can do all things through Christ" actually mean?

Verse 13 does not begin a new topic. It closes the paragraph that started at verse 10, where Paul thanks the Philippians for their gift and then explains that his thanks is not a complaint about need. He has learned contentment in plenty and in hunger, and verse 13 tells you how that was possible.

So the promise is enormous, but it is not a promise about outcomes. It is a promise about capacity: whatever state Paul is put in, Christ supplies strength sufficient to be content in it. That includes the states nobody chooses.

I have learned, verse 11, and I am instructed, verse 12. The second word carries the sense of being initiated or trained through experience. Paul did not arrive content. He was schooled in it, in both directions, and being full turns out to need teaching as much as being hungry does.

This is why the verse is a comfort rather than a rebuke. If contentment is learned, then a Christian who is not yet content is not disqualified. He is early in the course, and the teacher is the one who supplies the strength.

It will not carry the meaning that any goal a believer sets is achievable through Christ. Paul, using the same theology of strength, asked three times for a thorn to be removed and was refused, 2 Corinthians 12:8 to 9. He was strengthened, and he was not delivered.

Read honestly, the verse is more useful than the misquotation. A promise of unlimited success would fail the first time a Christian faced illness, bereavement or a closed door. A promise of sufficient strength in abundance and in need holds in every one of them.

The Philippians had supported Paul repeatedly. He states it himself: in Thessalonica they sent to his necessity more than once, verses 15 and 16, and they were the only church in partnership with him when he left Macedonia. This letter is in part a thank-you note, which is why money, need and contentment are the subject at the end of it.