If Galatians were summarized in a sentence, it could be this: justification is by faith alone and not by doing the works of the law (Gal 2:16; cf. 2:16–21). Supporting this truth in Galatians 3:10–14, Paul quotes the OT numerous times.
First, in Gal 3:10, Paul quotes Deut 27:26 and adds wording from Deut 28:58 as well. His point here is that, rather than finding justification, those who rely on the works of the law will actually find themselves cursed with death for failing to perfectly obey the law. Paul assumes here a truth so obvious that it does not even need stating—that, due to sin, it is impossible for anyone to perfectly obey the law (cf. 1 Kgs 8:46; Prov 20:9; Ecc 7:20). So, anyone who attempts to do the law in order to be justified actually ends up under the curse of death.
Second, in Gal 3:11, Paul quotes Hab 2:4. His point here is to again show that doing the law does not justify anyone. The quote positively shows that those who are righteous will live (eternally) by faith. The quote negatively implies that the law has no role to play in being justified before God because it says nothing about the matter and has already positively affirmed the role of faith. So, it is clear that no one is justified before God by the law.
Third, in Gal 3:12, Paul quotes Lev 18:5. His point is to show that the law grants life (theoretically) by simply doing it. Faith is not even mentioned. But again, here we assume that doing the law perfectly is impossible due to sin, which means that no one can ever attain eternal life thereby.
Fourth, in Gal 3:13, Paul quotes Deut 21:23. His point here is to show how Jesus became a curse for us so that He could die for our sins on our behalf. After an execution for some heinous sin, the body of the punished would sometimes be hung upon a tree as a public display of what happened to those who sinned in such a way (see Deut 21:22–23). While not yet dead, Jesus hung on the cross and then died on it, and the OT text could be applied to Him as well. He did nothing to deserve this curse and punishment, and what He suffered, He suffered for us.
There is obviously more to Gal 3:10–14 than what is said above, but we have at least seen that (1) all are under the curse of death, (2) righteousness comes by faith, (3) no one can do the law perfectly, and (4) Jesus took the curse of death upon Himself for us when He died on the cross. Praise God for a wonderful Savior!