If you run the vst() function on an object of class DESeqDataSet you get the results wrapped in a class called DESeqTransform, which basically a SummarizedExperiment (SE). A SE can store one or many count matrices which are called assays. To access an assay you can do (assuming your SE is called se):

assay(se, assayname)

If you only use assay(se) then the first assay is returned. Use assayNames(se) to list all available assays.

So the code does the following:

#/ this extracts the vsd assay (so the vst-corrected counts) from the vsd which is the DESeqTransform
#/ which is the first (and only) assay by default
mat <- assay(vsd)

#/ limma stuff...
mm <- model.matrix(~condition, colData(vsd))
mat <- limma::removeBatchEffect(mat, batch=vsd$batch, design=mm)

#/ this feeds the batch-corrected counts (=mat) back into the DESeqTransform object called vsd
assay(vsd) <- mat

The reason the last line of code above is done is that the plotPCA function works only with a DESeqTransform object, hence the mat is put back into the vsd object which is a DESeqTransform. This is not "assigning" mat to a function. Instead the assay command is a so-called setter function which is meant to put an object (here mat) into the correct location of the vsd. The assay serves both as a setter and also as a getter (so retrieval) function. This is a principle to conveniently put data into and extract from certain classes rather than having the user find these slots with access operators such as object$(...) or object@(...).

#/ now run PCA:
plotPCA(vsd)

Does that make sense to you?