dimnames() returns a list. The indexing of [[1]] returns the first element of the list. That first element is a vector. Appending [1] to the first element of the list returns the first element of the vector that is the first element of the list. Here is an example. dimnames(MAT) returns a list. The first element of that list is the vector c("A","B","C"). The first element of that vector is "A".
MAT <- matrix(1:9,nrow=3,dimnames = list(c("A","B","C"),c("Q","W","E")))
MAT
Q W E
A 1 4 7
B 2 5 8
C 3 6 9
dimnames(MAT)
[[1]]
[1] "A" "B" "C"
[[2]]
[1] "Q" "W" "E"
dimnames(MAT)[[1]][1]
[1] "A"