This verse has helped many a Christian through difficult situations. It is not an idle boast in one's self. Indeed, it is a boast in Christ Jesus, for without Him we are nothing, but with Him we are made strong enough to accomplish the tasks we are called to do. Not by our might, but by His Spirit, can we make it through the floods and fires of life.
The first part seems like a boast in our own strength, but the next two words in the verse take this out of the context of arrogance altogether: "through Christ." It is not a boast of self: "I can do all things." It is, I can do all things through Christ! There is a huge difference. It is one thing for someone to say, "I can do all things." It is quite different to confess, "I can do all things through Christ."
To get the impact of this, stop and think about what you couldn't do without Him! You could not effectively and thoroughly know your lost condition. Without Him, you could not have adequate concepts of God. Without Him, you could realize no victory over sin. Without Him, we would all be a victim of the wrath to come. Jesus said to His disciples "without Me you can do nothing," (John 15:5). In Ephesians 2:12 teaches outside of Christ, we have "no hope" and we are "without God in the world."
Sp with Christ, through Him, in our relationship of active faith in Him, we have strength we could not have any other way! Strength to adequately evaluate ourselves. Strength to know the difference between right and wrong (by faithfully applying His Word). Strength that comes through the remission of our sins. Strength in response to prayer we offer to God through Him. Strength of character as we follow His example and obey Him. Inner strength to make it through struggles. Physical strength to heal and recover.
This is strength in such abundance, we are able to do everything God wants us to do. "I can do all things through Christ, who strengthens me."
"Of the strength which Christ can impart, Paul had had abundant experience; and now his whole reliance was there. It was not in any native ability which he had; not in any vigor of body or of mind; not in any power which there was in his own resolutions; it was in the strength that he derived from the Redeemer. By that he was enabled to bear cold, fatigue, and hunger; by that, he met temptations and persecutions; and by that, he engaged in the performance of his arduous duties" (Barnes).
