I believe the humor behind this particular cartoon is that this man can no longer think by himself and is depending upon an Internet connection for him to perform the most basic brain functions such as "staying on".
While "me apago" could be correctly translated as "I am turning myself off" in this particular context is more like "I am powering off". When you turn on an electrical device you are "powering it on", when you turn it off you are "powering it off". As mentioned by Jdamian, this would be akin to "I am shutting down" which was what I was looking for when writing this answer.
"No tengo conexión" is frequently used to express "I have no Internet connection" or "I have no phone service".
So, I am assuming in this context he (or his brain) is expressing "I am powering off (I am shutting down), I have no internet connection". He is expressing this in an anxious manner, so this cartoon might be trying to make fun of how nowadays people can't do anything unless they are connected to the Internet because they can't think for themselves. It's obviously an exaggeration, of course.
If there is more to this cartoon (like other panels, a caption, a tool-tip, etc.) please share it, it might help to understand its meaning since the punchline could be hidden there.
Update: Thanks to fedorqui, it is clear that there is no hidden punchline for this cartoon. I had suspected the author was making fun of younger generations for their inability to do anything without an Internet connection but I refrained from adding such conclusion since there was no proof that the author would be of an older generation (I didn't know it was El Roto); however, it appears my suspicions were not far off.