When you see the "arranged by" attribution on a score, what did the arranger (typically) do?
An arrangement is about which instruments play what, when and how for a specific tune.
The core of a tune, or composition, is the melody/-ies1. Using this core an arranger - i.e. the person attributed by "arranged by" - when creating an arrangement may
- Decide what instruments, including singing voices, to use
- Select key (or keys) for the arrangement (i.e. the score or the performance)
- Decide on style (or styles) such as swing, bossa nova, baroque, raggae, etc
- Decide how to notate the music, including what meter to notate the music in (3/4, 6/8, 4/4,...)
- Indicate tempo(s)
- Decide on interpretation(s) of the melody/-ies in terms of
- rhythmical and tonal variations
- Decide the form of the music arrangement in terms of
- repeats and usages of the melodic units
- special choruses
- key changes/modulations
- tempo changes
- style changes
- meter changes
- intended climax
- Decide how to harmonize the melody
- Specify which instrument should play what and when, i.e. assigning the actual notes and/or roles to play within the decided form, including considerations of
- voicing and voice leading
- background voices
- tone color
All of this is notated in sheet music (as per your question regarding musical score), learned by ear by the musicians as in a "head arrangement", or created during recording sessions. The person(s) attributed as arranger has likely done some or all of the bullets above for the sheet music, or the musical production.
If the music in the score is (rather) exactly how the composer originally arranged it, then the "arranged by" attribution is generally omitted and it is understood that it (likely) is arranged by the composer or (rather much) as the composer once did or intended.
Arranging calls for a lot of skills including
- Instrument knowledge including posibilities and limitations
- Orchestration knowledge
- Musical style knowledge
- Music theory especially including harmony and different arranging styles
- A musical ear
1For some music there may be other aspects than a melody that constitutes the core of a tune, however I believe some or most of this answer will still apply.
An arranger specifically changes the music away from what was originally written. This is different from an editor who might clean up notation, clarify the meaning of markings, translate outdated terms, give instructions for proper interpretation, etc. Both technically alter the original but the editor tries to "bring out" the original as much as possible, whereas the arranger does ... something else.
An arrangement is often about simplification; you'll often see this sort of arrangement in beginner's books. The key could be changed to C to keep everything in the bigger white keys on a piano, or the chords transposed to make it easier to play on a guitar or possible to play on a fixed-key instrument, for example. Other simplifications could include removal of trills, reducing the number of notes played (whether by decreasing note frequency or the complexity of chords), and so on.
Other arrangements are about reinterpretation of a piece. I have an arrangement of Canon in D for solo piano (it is originally for 3 violins and basso continuo). Another option would be changing the style, such as arranging a jazz cover of a punk rock song :P. One more case I've seen is where someone transcribes a song by ear. Since such transcription is unlikely to be perfect, it would be considered an interpretation.
As such, there is a lot you can do with arrangements. You can't really assume anything when you see "arranged by..." without knowing something about the original, other than that something has changed. The Wikipedia article on arrangement has a couple good quotes in the introduction that capture this idea.