The ESV Study Bible explains:
1 Tim. 4:10 to this end. The goal of Paul’s labors is that people attain “godliness” (v. 8) and its eternal “value.” Toil and strive is typical of Paul’s description of gospel ministry (cf. 5:17; Rom. 16:6, 12; 1 Cor. 15:10; 16:16; Gal. 4:11; Eph. 4:28). The statement that God is the Savior of all people, especially of those who believe could seem to teach UNIVERSALISM, that every person will eventually go to heaven. However, the rest of Scripture clearly denies this idea (see note on 1 Tim. 2:4). There are several other possible explanations for this phrase: (1) It means that Christ died for all people, but only those who believe in him are saved. (2) It means he is offered to all people, though not all receive him. (3) It means “the Savior of all people, namely, those who believe” (a different translation of Gk. malista, based on extrabiblical examples). (4) It means “the helper of all people,” taking Greek Sōtēr, “Savior,” to refer not to forgiveness of sins but to God’s common grace by which God helps and protects people in need. (5) It means “the Savior of all kinds of people, not Jews only but both Jews and Greeks.” In any case, the emphasis is on God’s care for the unsaved world, and in the flow of the letter Paul is stressing once more (cf. 2:3–5) that God’s will that people would be saved is the basis of the universal mission (cf. Matt. 28:19–20). On God as “Savior,” see note on 2 Tim. 1:8–10.
The ESV MacArthur Study Bible provides a commentary about this verse:
1 Tim. 4:10 hope. Believers are saved in hope and live and serve in light of that hope of eternal life (Titus 1:2; 3:7; see note on Rom. 5:2). Working to the point of exhaustion and suffering rejection and persecution are acceptable because believers understand they are doing God’s work—which is the work of salvation. That makes it worth all of the sacrifices (Phil. 1:12–18, 27–30; 2:17; Col. 1:24–25; 2 Tim. 1:6–12; 2:3–4, 9–10; 4:5–8). the Savior of all people, especially of those who believe. Paul is obviously not teaching UNIVERSALISM, that all people will be saved in the spiritual and eternal sense, since the rest of Scripture clearly teaches that God will not save everyone. Most will reject him and spend eternity in hell (Matt. 25:41, 46; Rev. 20:11–15). Yet, the Greek word translated “especially” must mean that all people enjoy God’s salvation in some way like those who believe enjoy his salvation. The simple explanation is that God is the Savior of all people, only in a temporal sense, while of believers in an eternal sense. Paul’s point is that while God graciously delivers believers from sin’s condemnation and penalty because he was their substitute (2 Cor. 5:21), all people experience some earthly benefits from the goodness of God. Those benefits are: 1) common grace—a term that describes God’s goodness shown to all mankind universally (Ps. 145:9) in restraining sin (Rom. 2:15) and judgment (Rom. 2:3–6), maintaining order in society through government (Rom. 13:1–5), enabling man to appreciate beauty and goodness (Ps. 50:2), and showering him with temporal blessings (Matt. 5:45; Acts 14:15–17; 17:25); 2) compassion—the broken-hearted, loving pity that God shows to undeserving,......