It's not a very well constructed sentence.
The first part is fine. The implied clause "(human beings) claim a God" is slightly odd use of claim, but I can understand it.
Human beings are only creatures on Earth that claim a God
the second part (after and) means
[human beings are] the only living thing that behaves like it has not got [a God]
Which is uncomfortable as it shifts between the plural and the singular. You can defend it by understanding a "living thing" to refer to a species, so the plural (human beings) is one species (humanity) but it could probably be rewritten.
The suggestion is that other living things (cats, mosquitos, trees, bacteria) don't say that God exists, but other living things behave as if they are following the will of God.