This could be us, but I hate running and you don't exist.
Can't understand the meaning when translated into Russian language. Please explain how a native English speaker, you understand the phrase.
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The meaning is literal. It's supposed to be funny. This could be us implies that the speaker and another person could be like the pair in the picture. You can infer from the image that the speaker is saying that "we" could be running and happy. The speaker is expressing this possibility.
You can interpret the bottom portion as the "truth". The truth is that "I" hate running. You don't exist could be interpreted in a few different ways. One is that you in fact don't have a boyfriend/girlfriend or friend you could run with. Another is that your ideal girlfriend/boyfriend or ideal friend doesn't exist. Regardless, the truth is "you" don't exist. These facts contradict the possibility of "us" being the pair in the picture. It was never actually possible. I believe this is called "misdirection" in comedy terminology.