Since we have heard -- The brethren at Jerusalem first heard about the Judaizers at Antioch when the delegation arrived in Jerusalem from Antioch.
that some of our number -- The Judaizers were "members" (see Acts 15:1) of the Jerusalem congregation before they undertook their trip to Antioch.
They had been unaware of some Jerusalem members traveling to Antioch who demanded conformity to the Mosaic Law.
to whom we gave no instruction -- These Judaizers had no authority from the Jerusalem church, though they evidently had been claiming to have come from the mother church, sent by them on an official mission to Antioch.
The opening paragraph of the letter sent to Antioch is an official denial that the Judaizers had ever been authorized or sent by the church at Jerusalem. We cannot help but wonder if the Judaizers were present in the assemble at Antioch when this epistle was read to the congregation, and if so, what their reaction was. - Gareth L. Reese
have disturbed you with their words -- The word "disturbed" has a number of connotations. Literally, it is used of waters that are "troubled." In a figurative sense the word means to stir up, disturb, throw into confusion, be agitated, or intimidated.
words -- "Words" here might have the connotation that they were mere words, words without true doctrine. They were just words. They weren’t the truth. Nevertheless, such words can have a very serious unsettling effect on men’s minds. - GR
unsettling your souls -- This verb (ἀνασκευάζοντες, anaskeuazontes) occurs nowhere else in the N.T. In classical Greek it meant "to collect together the vessels used in a house- the household furniture and silver - for the purpose of carrying it off." Then it meant to dismantle, to overthrow. In a military connotation it was used for plundering (laying waste, or ravaging) a city.
Here the idea seems that of turning the minds of the Gentile converts upside down; throwing them into confusion like a dismantled house.
Metaphorically it can mean "to break a treaty" or "to destroy an opponent’s arguments."
Whenever a man’s salvation is called in question, and it appears that the one who has asked the questions have been teaching doctrines (the verb tense here is present tense -- they might still be at it) which unsettled the minds of the Christians, and left them disturbed and anxious.
saying "You must be circumcised and keep the law" -- This phrase is omitted from most uncial manuscripts, but found in almost all miniscule manuscripts. The NRSV places this phrase in a footnote saying, "Other ancient authorities add saying, ‘You must be circumcised and keep the law,’"
Most early manuscripts end the verse with “upsetting your minds,” but some witnesses also include “saying, ‘You must be circumcised and keep the law.’ ” The presence of this text explicitly provides the material that has confused the letter recipients. - The Lexham Textual Notes on the Bible
24 ἐξελθόντες Treg NA28 RP ] – WH
[RP: The New Testament in the Original Greek: Byzantine Textform 2005, compiled and arranged by Maurice A. Robinson and William G. Pierpont (Southborough, Mass.: Chilton, 2005).]
[Apparatus for the Greek New Testament: SBL Edition]
1) While most uncial MSS omit this phrase, it can be deduced from Acts 15:1 that the apostles and Jerusalem elders did not sanction the teaching that unless the Gentiles were circumcised according to the law of Moses they could not be saved.
2) It can be observed from Acts 15:5 and this verse that the apostles and Jerusalem elders did not charge Gentiles converts to "keep the law of Moses."
Consider (Acts 15:9-10; Galatians 2:3-4; Galatians 6:12-13).
By teaching "circumcision" the Judaizers engaged the observance of the whole law of Moses.
to whom we gave no such commandment -- The Judaizers were mis-representing the Jerusalem church, they did not have such authority from the apostles or Jerusalem elders as they seemed to have claimed (Acts 15:1-2). (1 John 2:19)
The point is then that the apostles (and Jerusalem elders) had given no charge (orders) on the points which the Judaizers had raised.
15:24 This verse shows that the church in Jerusalem had become aware that some of their membership, who had no authority or official standing (cf. v. 1), were (1) traveling to these mission churches and (2) demanding conformity to the Mosaic law (cf. v. 1). The VERB (anaskeuazō) used is a strong military term used only here in the NT for plundering a city. - Utley Bible Commentary
On the fundamental matter of the theological necessity of circumcision and a Jewish lifestyle for Gentile Christians, the letter rebukes the Judaizers for going beyond their authority and assures the churches that there are no such requirements for salvation. On the practical issues of fellowship between Jewish and Gentile believers in the churches and of preventing needless offense to Jews throughout the empire, the letter asks Gentile Christians to abstain from the four prohibitions mentioned in v.20 - THE EXPOSITOR’S BIBLE COMMENTARY, New Testament. Kenneth L. Barker, John R. Kohlenberger III; Zondervan
[On the differences between the Old Uncial and the Western textual readings, see C. K. Barrett, “The Apostolic Decree of Acts 15:29, ” Austrialian Biblical Review 35 (1987):50-59.]