It means that they have never personally experienced whatever particular situation you were talking, but they are able to guess that their reaction to it would've been the same as yours. Example; "I'm so exhausted from working 70 hours this week." "I can only imagine."

It is often used in a situation where something causes a reaction and we are trying to say that more of the cause would result in a reaction that is hard to estimate. "If he gets that mad over little things, one can only imagine the explosion if something really bad happens."