http://analytics.ncsu.edu/sesug/2014/SD-13.pdf

It provides the following OR interpretation of a continuous fixed effects covariate at level 1 in the model:

From the OR output (details not shown) it appears that for every one unit increase in student SES the odds of passing the exam increase by 1.76. That is, for students with personal SES one standard deviation above the mean, their odds of passing the math test is 1.76 times greater than students with average SES.

I always interpreted a 1 unit increase as a 1 unit increase in the continuous variable, not a 1 standard deviation increase. So if the continuous variable was age, a one unit increase would be 1 more year not a standard deviation increase.

Historically for me this has been in the context of logistic regression not MLM, but I don't think it should matter.