A lot of things come into play here. Often, the harmonies are analyzed (using Roman Numerals if it's Common Practice Tonality, using something like set theory if it's post-tonal, etc.) in order to see where the composer has followed standard progressions and, more interestingly, when they haven't. We look at the specific voicings of the harmonies and see how that influences the overall sound. In common-practice music, we look at the specific ways the melody interacts with the harmony, where are non-chord tones used, do particular patterns emerge, etc. We look for the ways that particular motives get mutated, copied or adapted.
Depending on the instrumentation, we often spend a lot of time on specific orchestrational choices by the composer. What's the effect of choosing one instrument or another? Would another instrument have been more traditionally suited to the material, is the standard range being stretched, etc. If the piece is serial, what is the tone row, and how does it manifest in vastly different situations across the piece? In what ways does it retain similarities despite these differences?
Do certain rhythmic, harmonic, motivic or timbral patterns recur in later situations? How are they similar and how are they different?
There are many, many other things that are specific to particular music. Score study of a fugue is very different from score study of a Bartokian exploration of pitch-space symmetry, but ultimately I'd say that the main benefit of score study is that you notice things in the pieces that you haven't noticed before. I'm often surprised by how much I learn about a piece I thought I already knew well even by just following along with the score before I've even done any actual analysis. If you can get a book that contains scores alongside analytical commentary that can help, or if you have access to JSTOR or something similar you could look up analyses of pieces you know and have scores to.