I can't (using the example code from MCMCglmm) construct an example where as.data.frame(model$Sol) gives me a dataframe of dataframes. So although there's probably a simple answer I can't check it very easily.
That said, here's an example that might help. Note that if your child dataframes don't have the same colnames then this won't work.
# create a nested data.frame example to work on
a.df <- data.frame(c1=runif(10),c2=runif(10))
b.df <- data.frame(c1=runif(10),c2=runif(10))
full.df <- data.frame(1:10)
full.df$a <- a.df
full.df$b <- b.df
full.df <- full.df[,c("a","b")]
# the solution
res <- do.call(rbind,full.df)
EDIT
Okay, using your new example,
require(MCMCglmm)
data(PlodiaPO)
result<- MCMCglmm(PO ~ plate + FSfamily, data=PlodiaPO,nitt=50,thin=2,burn=10,verbose=FALSE)
melt(do.call(rbind,(as.data.frame(result$Sol))))