Do anyone know a R function (or another way) to get the proportion of explained variance by each one of the fixed effect factors in a linear mixed-effects model?
This is actually a very complex question. Defining what the proportion of explained variance means in these models is a non-trivial exercise. I would start with chapter 7 of Tom Snijders & Roel Bosker (1999) "Multilevel Analysis: An introduction to basic and advanced multilevel modeling" Thousand Oaks: Sage.
Nakagawa and Schielzeth (2013) has become the standard approach to this problem (currently cited almost 2000 times). See R2 for mixed models with multiple fixed and random effects.
Note, however, that the encyclopedic GLMM FAQ highlights that this is a complex issue and does not outright endorse the Nakagawa and Schielzeth approach, instead highlighting it among various other approaches and emphasizing that
'you have to think carefully about what information you want to get out of the coefficient of determination’, because no recipe will have all of the properties of R2 in the simple linear model case.