3 As I urged you when I went into Macedonia—remain in Ephesus that you may charge some that they teach no other doctrine,
4 nor give heed to fables and endless genealogies, which cause disputes rather than godly edification which is in faith.
urged (v.3) = exhorted, encouraged, begged. This occurred sometime before this letter was written.
remain (v.3) — apparently, Paul and Timothy had been at Ephesus together. Paul left for Macedonia, but encouraged Timothy to remain and then wrote this letter to give him in writing what had been verbal instructions.
other (v.3) — The verb heterodidaskaleo, to teach something different, is used only here and at 6:3, in the Greek Bible. Compare to the word “different” (heteros) in Galatians 1:6, followed by allow, “another of the same sort,” in verse 7. — Vine, page 143
The words, “teach no other doctrine,” are the translation of a negative and the verb heterodidaskaleo, the latter word being made up of the verb, “to teach,” and the word heteros, which means, “another of a different kind.” The idea is not merely that those exhorted are not to teach any other doctrine, but they are not to teach a different doctrine, a doctrine which is contrary to the true doctrine. Our word, “heterodoxy,” namely, false doctrine, refers, not to doctrines of false religions, but to doctrine which poses as true Christian doctrine, but which is diametrically opposed to the true teachings of Christianity. This is what Paul is referring to. — Wuest, page 26.
charge some (v.3) — take a strong stand
give heed (v.4) = attach oneself to, apply oneself to, cleave to — not just listening, but attaching oneself to the teachers, hold to, give assent — In this case, Paul was telling Timothy NOT to do this
fables (v.4) = myths — possibly legendary Jewish or Gnostic stories of history
endless (v.4) = without limit, without resolution and, therefore, useless
genealogies (v.4) — fabricated or speculative
cause (v.4) = provide for, give occasion to
disputes (v.4) = questionings, investigation, laborious inquiry
edification (v.4) = dispensation — teaching about God’s stewardship, His dealing with the world and humanity
in faith (v.4) — taking God at His word rather than relying on those who claim to be wise
Paul predicted all this in Acts 20:29-30: For I know this, that after my departure savage wolves will come in among you, not sparing the flock. Also from among yourselves men will rise up, speaking perverse things, to draw away the disciples after themselves.
Wuest’s translation — Just as I begged you to continue on in Ephesus, when I was going into Macedonia, in order that you might charge certain ones not to be teaching things contrary to sound doctrine, nor to be giving assent to fables and useless genealogies which are of such a character as to provide occasion for exhaustive investigations rather than a (knowledge of the) administration of the things by which God has provided for and prepared salvation, which salvation must be embraced by faith. — Wuest, page 28.
My take on these verses is that Paul was telling Timothy to stand firmly against those who professed to know more about doctrine than what God clearly states in His Word. Rather, by simple faith, take God at His Word and you will understand His stewardship. So, speak against those who claim that science can give us greater insight into creation than Genesis, or those who give weight to fables about kids who go to heaven or people in foreign countries who see apparitions of angels and ignore what the Bible actually says.