What -n and -p do is wrap something like the following around your code (no, it's not the exact code; just a rough facsimile)

while( <> ) {
  # -e goes here
} continue {
  print;
}
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If you have a BEGIN {} or END {} in awk the code in those is run before or after the first line of input is read. So it's in effect saying you get:

## code that was in BEGIN block
while( <> ) {
  ## -e code
} continue {
  print;
}
## code that was in END block
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