Would someone under the effects of comprehend languages understand Thieves' Cant? Specifically, would they understand the meaning of what the person was saying and not just the words?
Comprehend languages only lets you understand the literal meaning of the words.
For the duration, you understand the literal meaning of any spoken language that you hear.
Thieves cant states that the message is coded so only thieves would understand.
During your rogue training you learned thieves' cant, a secret mix of dialect, jargon and code that allows you to hide messages in seemingly normal conversation.
So while you may understand the literal message, "The rooster flies when the blood moon rises," you will not understand that this means that, "The man with the red scar on his neck is your target for assassination by midnight tonight."
This is akin to telling somebody to read the 3rd letter of the first paragraph in a book, in which it spells out a secret message to them. Comprehend languages would let you understand the paragraph, but would give you no insight whatsoever to the hidden code inside it.