Exegesis:

hoi de epischuon legontes ‘but they insisted saying,’ i.e. ‘by saying.’

epischuō lit. ‘to be strong,’ here ‘to insist,’ ‘to persist.’

anaseiei ton laon ‘he stirs up the people.’ For laos cf. on 1.10.

anaseiō ‘to stir up,’ ‘to incite,’ probably somewhat stronger than diastrephō in v. 2.

didaskōn kath’ holēs tēs Ioudaias ‘teaching throughout all Judea.’ didaskōn may mean ‘while teaching’ (temporal) or ‘by teaching’ (causal), preferably the latter. Ioudaia is used here to indicate the area of Pilate’s jurisdiction.

kai arxamenos apo tēs Galilaias heōs hōde lit. ‘and having begun from Galilee till here,’ attached to the indication of Jesus’ present area of activity in order to indicate the origin of his operations. It is best rendered as an independent clause.

Translation:

They were urgent. The pronoun may have to be specified, cf. e.g. ‘the accusers’ (Javanese). The verb expresses that they firmly and stubbornly kept on accusing Jesus, repeating and pressing their charge against him, cf. ‘they held yet more determinedly to their earnest demand’ (Marathi), ‘they kept on pressing (the matter)’ (Zarma), ‘they repeated-stubbornly’ (Pohnpeian).

The people, cf. on 7.29.

Teaching, preferably, ‘by, or, with his teaching,’ ‘because he teaches them these/such things.’

Throughout all Judea, cf. on 7.17 and 4.44.

From Galilee even to this place, or as a new sentence, “He started in G. and now he has come here” (The Four Gospels – a New Translation, similarly An American Translation), “It (referring to Jesus’ teaching and to the disaffection it causes) started from G. and has spread as far as this city” (New English Bible).

Quoted with permission from Reiling, J. and Swellengrebel, J.L. A Handbook on the Gospel of Luke. (UBS Handbook Series). New York: UBS, 1971. For this and other handbooks for translators see here . Make sure to also consult the Handbook on the Gospel of Mark for parallel or similar verses.