This is the 1183rd most frequent Portuguese word.
conseguiram
"They managed" or "they succeeded."
The word 'conseguiram' indicates that 'they' (third person plural) successfully carried out the act of solving the problem.
Eles finalmente conseguiram resolver o problema.
They finally managed to solve the problem.
Here, 'conseguiram' expresses that 'the children' successfully completed the task of assembling the puzzle after trying.
Depois de muito esforço, as crianças conseguiram montar o quebra-cabeça.
After much effort, the children managed to assemble the puzzle.
In this question, 'conseguiram' is used to ask if 'you all' (second person plural) successfully understood the explanation provided by the teacher.