When my cousin was only 1 years old, my aunt texted me a video where she was crying.
= When my cousin was only 1 year old, my aunt sent me a video of her crying.
 

She cried and said "Show me my video!", but no tears came out!
= She cried and said, “Show me my video!”, but no tears came out.
 

It turns out, she just wanted to watch a video which she was taken and cried for let her mother know what she wants.
= It turned out that she just wanted to watch a video that had been taken of her, and she cried to let her mother know what she wanted.
 

And her mother showed the video to her, she stopped crying as noting was happened.
= When her mother showed her the video, she stopped crying as if nothing had happened.
 

It was so funny and I realized that a baby can cry crocodile tears.
= It was so funny, and I realized that even a baby can cry crocodile tears.