How are Scala collections able to return the correct collection type from a map operation?

Note: This is an FAQ, asked specifically so I can answer it myself, as this issue seems to come up fairly often and I want to put it in a location where it can (hopefully) be easily found via a search

As prompted by a comment on my answer here

For example:

"abcde" map {_.toUpperCase} //returns a String
"abcde" map {_.toInt} // returns an IndexedSeq[Int]
BitSet(1,2,3,4) map {2*} // returns a BitSet
BitSet(1,2,3,4) map {_.toString} // returns a Set[String]

Looking in the scaladoc, all of these use the map operation inherited from TraversableLike, so how come it's always able to return the most specific valid collection? Even String, which provides map via an implicit conversion.