What does "I plead the blood of Jesus Christ" mean?

It's not an expression I hear, but "the blood of Jesus Christ" is New Testament shorthand for "the fact that Jesus died". So "pleading" it would mean "I rest my defence on the fact that Jesus died". This would make it a metaphorical legal term, referring to the "case" in which we stand before God in judgment.

It would be another way of saying "We have an advocate with the Father, Jesus Christ the righteous" (1 John ch2 v1).

The saying could be said, in private prayer perhaps, by individual Christians, in speaking to God against whom they have sinned.

(Against thee, thee only, have I sinned ... Psalm 51:4.)

If so, it would probably not be reported publicly, but would only be a private communication betwixt the individual and God.

There are many texts in scripture which would support such an utterance :

... Christ Jesus, whom God hath set forth a propitiation through faith in his blood [Romans 3:25]

... being now justified by his blood ... [Romans 5:9]

... in whom we have redemption through his blood ... [Ephesians 1:7]

... How much more shall the blood of Christ ... purge your conscience ... [Hebrews 9:14]

... Jesus, the mediator of the new covenant, and to the blood of sprinkling, that speaketh better things than that of Abel ... [Hebrews 1:24]

... Unto him that loved us and washed us from our sins in his own blood ... [Revelation 1:5]

All quotations are from the Authorised, King James, Version.

I have linked this answer to an article written by an American televangelist named Kenneth Copeland. It is an instructional article entitled "How To Plead the Blood of Jesus" and it explains not only how but why one may "plead the blood". Mr. Copeland is associated with the Charismatic Word of Faith movement which is known to put an undue emphasis on procuring health and wealth through the gospel. Here is a link to more information on what is sometimes called "name it and claim it" Christianity.

I do not biblically agree with the great majority of the Word of Faith teachings nor how they make application of biblical principles, however the principles themselves are sometimes accurate. I believe the notion of 'pleading the blood' is one of those principles, correct but misused and misapplied.

Mr. Copelands premise of "pleading the blood of Jesus" is rooted in the fact that believers have been purchased out of every kindred, and tongue, and people, and nation by the blood of Jesus Christ (Rev 5:9). There are a great many passages in Scripture which buttress this teaching of believers having been purchased; among them are 1 Peter 1:18-19, 1 Corinthians 6:19-20, and Acts 20:28:

Take heed therefore unto yourselves, and to all the flock, over the which the Holy Ghost hath made you overseers, to feed the church of God, which he hath purchased with his own blood.

Mr Copeland's article indicates that there is no longer any legal claim that Satan can make to the one who is in Christ, who has been purchased or redeemed from the slave market of sin with His own precious blood:

When you plead the blood, you have laid down your case and put your entire confidence on an oath that is covenant-sworn by Almighty God.

When a believer sins (and we all do) we may have great confidence in approaching God for forgiveness since we know that the shed blood of Jesus is always efficacious when repentance is genuine. Even when our consciences are weighed down by guilt and the enemy of our souls stands accusing (and rightly so when we sin) his legal claim disappears when we appeal for forgiveness on the basis of Christ's shed blood. From God's standpoint the sin is already forgiven, already taken away, and this already procured forgiveness is accessed through repentance and faith. The legal claim of the law is nullified and the principalities and powers which stand against us are without any claim:

And you, being dead in your sins and the uncircumcision of your flesh, hath he quickened together with him, having forgiven you all trespasses; Blotting out the handwriting of ordinances that was against us, which was contrary to us, and took it out of the way, nailing it to his cross; And having spoiled principalities and powers, he made a shew of them openly, triumphing over them in it. - Colossians 2:13-15

All that being said, Mr. Copeland takes the principle much, much too far; insisting that sickness, disease, poverty, lack, depression, strife, etc., can be "pled" away. Indeed, he would have us "pleading the blood" nearly constantly as though it were some kind of magical incantation:

You can plead the blood of Jesus over any person or situation. As your children walk out the door, lay your hands on their heads and say, “In the Name of Jesus, I plead the blood of Jesus over [name].” As you drive, say, “In the Name of Jesus, I plead the blood of Jesus over this vehicle and all who are in it. We will go and come back in perfect safety.”

We may certainly bring all of our requests and petitions to the Lord but there is no formula nor phraseology which we are given to wield with power. In this, Mr. Copeland is dreadfully wrong. In contradiction to this thought process the Scriptures encourage us to patiently endure and rejoice in all manner of trials, not plead them away.

Count it all joy, my brothers, when you meet trials of various kinds, for you know that the testing of your faith produces steadfastness. - James 1:2-3

I will plead the blood of Jesus in repentance as I petition for forgiveness of my sin (as there is no other recourse but that precious redemption in His blood) but I will not demean the sacrifice by naming it in pursuit of creature comfort as though the Lord is not sovereign over my trials.