Context is critical. The Corinthians had created factions among themselves, and set up conflicts between who was a follower of Paul, who was a follower of Apollos, etc. and created division and strife within the church (see chapter 3). Which meant that some had made judgments that other ministers were either superior or inferior to Paul. They were carnally judging the merits of the servants of the Lord and becoming followers of men, instead of promoting unity and giving all the same honor and respect. Therefore Paul said: Let a man so account of us, as of the ministers of Christ, and stewards of the mysteries of God.
It is in this context that Paul tells them to judge nothing before its time. Many Christians were in the habit of underestimating Paul, because he did not have an impressive appearance and his speech was not oratorical. He also had a physical disability (either with his legs or his eyes).
Others were opposing Paul because of his faithful Christian doctrine (as opposed to the Judaizers). And Paul had to constantly assert his apostleship, and his apostolic authority. Today we have some who brazenly claim that Paul was not a true apostle and that his writings should be ignored.
This passage is not telling Christians to stop making moral or spiritual judgments, since Christ already said that we are to make righteous judgments, and Paul said that we are to prove all things, and hold fast to that which is good. The apostle John said that we are to try or test the spirits, and the NT tells Christians to identify the wolves among the sheep. At the same time Christians are not to judge the spiritual standing of others, since only God knows hearts. Some may be saved, and yet be carnal, as were many of the Corinthians. However, "by their fruits ye shall know them" is still applicable.