They are stoning him, and as such take him outside the city to a pit. There, they will strip him and hurl rocks on him until he dies. They are to aim for the chest, but precision is impossible.
Under Jewish law,* the criminal was to be stripped (Mishnah Sanhedrin 6:4), but here the executioners strip themselves. The obvious reasons are that it was hot and/or they didn't want to get blood on their clothes (as Affable Geek suggested).
*The citation is from the Talmud but the Mishnah portion. That makes it part of the older tradition that makes up the Talmuds. The Talmuds (Babylonian and Jerusalem) are made up of a shared Mishnah and differing Gemerahs.
As these are false witnesses, they are guilty and to be stoned according to Old Testament, Intertestamental, and Rabbinic law (Deut. 20:18-19; Susanna 55, 59, 62; Mishnah Makkot 1:6; Sanhedrin 11:6; Tosefta Sanhedrin 6:5; 9:5; 14:17; Sifre to Deuteronomy 190:4-5).
But why does Luke mention it? It seems like such a minor detail. He does introduce Paul/Saul here, and by keeping watch on the clothes, we see that Saul is a trusted individual. We also see something else: the accusers are acting decidedly unJewish. One of the reasons for the Hasmonean revolt was the influence of Hellenism--Greek living and ways of acting. Greeks exercised naked (that's where the word gymansium comes from, "to exercise naked"). 1 Maccabees begins with a recounting of how far the Jews have already been Hellenized. One of the first in the list is the building of gymnasiums in Jerusalem.
13 and a number of the people eagerly approached the king, who authorised them to practise the gentiles' observances.
14 So they built a gymnasium in Jerusalem, such as the gentiles have,
15 disguised their circumcision, and abandoned the holy covenant, submitting to gentile rule as willing slaves of impiety.
The conflict between Greek Jews and Hebrew Jews appears in other parts of Acts. Most notably, 6:1 and 9:29.
Acts 6:1 Now at this time while the disciples were increasing in number, a complaint arose on the part of the Hellenistic Jews against the native Hebrews, because their widows were being overlooked in the daily serving of food. [NASB]
When the first deacons are selected here in Acts 6, some will note that all seven are listed by their Greek names. This makes perfect sense as it was the Greek-speaking Jewish-Christians who felt they were being left out.
Acts 9:27 But Barnabas took hold of him and brought him to the apostles and described to them how he had seen the Lord on the road, and that He had talked to him, and how at Damascus he had spoken out boldly in the name of Jesus. 28 And he was with them, moving about freely in Jerusalem, speaking out boldly in the name of the Lord. 29 And he was talking and arguing with the Hellenistic Jews; but they were attempting to put him to death. [NASB]
Paul picks up right where Stephen left off, with the same kind of Hellenized Jew, and is met with the same results--they try to stone him.
Luke also has a purpose in his writing to show that Christianity is a branch of Judaism. If he can establish such, Christianity will be seen as a legal religion and not a cult. Showing this through detail instead of simply saying it was first the Greek-speaking Jews who had problems with Christians helps that argument. Legal religions were those which had a long history with the empire and were deemed safe. Judaism and Rome had enjoyed some level of relations since the time of 1 Macc 8:17-32. The times when Jewish and Roman relations would stretch, the revolt of AD 66-73,* the Diaspora Revolt of AD 114, and Bar Cochba's Revolt of AD 132-135 had not yet happened. If Luke could convince his audience that Christianity was connected to and a growth of the old Judaism, Christians would have the same rights and protection under the law as Judaism.
*I date Luke-Acts pre-64, but that would be an answer to a different question.
But I believe Luke has another reason for mentioning it beyond showing the Hellenization of the accusers. He is showing the guilty party. As I said, the law required the criminal to be stripped. Stephen probably was stripped, but Luke doesn't mention it. He mentions the accusers strip themselves in a bit of non-humorous irony--to show who the guilty party really is.