"经过/在....后” is equal to English "After+sb+did something ". For example, “经过/在他考虑(之）后，他决定去这家公司上班”. And in your case, "了解" shouldn't be translated as "understand", I think this is the part where you find it difficult to understand. “了解” here is more like "managing to know the whole story(The man was guilty and should be sentenced)".

For my explanation, here 经过 means Pass through (undergo). your understanding is right. After the police learn about it, the man was sentenced for 5 days in custody. but you translated the sentence not into the right word. but based in my English level, I couldn't give an accurate sentence, I try to say" After undergoing the police investigated, the man was sentenced for 5 days in custody. " 经过 can means go through a place or a gate, via a mall,via television or pass through a situation/procedure. undergoing some events. etc. but in chinese meaning, it is a verb and preposition that translates as after or go through and by.... I googled it that said it means that. I just do my best to clarify it. if you have any doubt please let me know.

“经过”=through。这里不是“经过”一个地方，而是“经过”一个procedure.

After the process of asking (many people) about the case (here, police is the logic subject) or after the process of carefully asking the man. (here, police is the logic subject) or after the process of doing all necessary work to fully understand what happened, (here, police is the logic subject) .... + the man was .... (the result)

Thanks, Darker. I get it now.