If you recognize that sign languages such as ASL are distinct languages, as linguists do, then it naturally makes sense that native speakers of those languages would want a way to write them down in a static symbolic way, for all the same practical reasons that we use the Latin alphabet in English.

For instance, being able to quickly scan through a piece of text instead of having to watch it play in video form, or being able to search and index it, or providing a way to organize dictionaries.

There's no inherent problem with using the same notation scheme for different sign languages, just like we use essentially the same alphabet for English, Spanish, French, German, etc.

It's a way to write sign languages. Think of like the alphabet, but for hands, movements etc instead of sounds.

Now, it may not be obvious that there is a necessity for a writing system for a minority language embedded in a larger community (spoken language), but there are many uses: preservation, digital use, teaching, linguistic study...

SignWriting is closer in purpose to the International Phonetic Alphabet for spoken languages. It attempts to allow detailed recording of the actual signing as it is signed for any sign language.

It has a lot of the disdvantages of IPA as a practical writing system as well.

Sign languages are not the same as spoken languages used in the same countries, as is very apparent if you look at transliterations of ASL using latin glyphs, there are some standardized ways to do this but they drop a lot of information and don't have the same sentence/word structure.

There is also a long history of attempts to create notation that can record this type of language, the first for ASL being Stokoe notation, which represents hand shapes for example, but can't represent for example facial elements, and is specific to ASL, can't represent things in other sign languages.

I don't have a good answer to this question at all, since I know barely anything about this.

However, something to keep in mind is the following: signed languages are not a signed transliteration of the local language. For example, American Sign Language is not a signed way to communicate English. It has its own grammar. Therefore, when you serialize something like ASL, you do not get back something like English.

So, you have to have a different way to serialize ASL, and this is that.

When I was learning BSL (British Sign Language), I wanted to be able to note down words I had learned, or the ability to look them up. It was part of how I studied. At the time, all that was available was https://en.wikipedia.org/wiki/Stokoe_notation which is a complete pain to try to use. Sutton SignWriting looks to be similar in that sense. As others have said, much like IPA for language learning.

Of course video is more easily available now, so there are some aspects of study that may not need the ability to write things down.