How to interpreting the exponential coefficent in poisson regression with offset?
The offset means "exposure", so when you are calculating a regression with offset, instead of calculating absolute values, you are calculating ratios.

You are changing "number of thefts" for "number of thefts for every 10.000 vehicles"

$y = x1+x2+(1|cluster) + log(x_3)$: This is the equation you are calculating.

I think you are missing a log in $y$ because the offset and the endogenous variable have to form a ratio:

$log(y) = x1 + x2+(1|cluster)+log(x_3)$

$log(y)-log(x_3) = x1 + x2+(1|cluster)$

$log(\frac{y}{x_3}) = x1 + x2+(1|cluster)$

$log(ratio) = x1 + x2+(1|cluster)$