The verse in the ESV reads
More in number than the hairs of my head
are those who hate me without cause;
mighty are those who would destroy me,
those who attack me with lies.
What I did not steal
must I now restore? (Psalm 69:4)
The context is certainly Messianic, with the verse being cited in the New Testament in relation to Jesus (John 15:25). However, the following verse is clearly not Messianic (“O God, you know my folly; the wrongs I have done are not hidden from you.”), and applies only to the Psalmist. So it is not certain that the last part of v4 is Messianic, although I think it is likely.
It appears that the Messiah’s enemies falsely accused him and held him accountable for crimes he did not commit. We know they falsely accused Jesus of many things (Matthew 26:60-61) although there is no specific reference to an accusation of robbery. But perhaps it is a proverbial expression meaning “I must suffer for things I haven’t done”.