Jonathan has given you a good explanation about the spelling, but I presume that what you are actually asking is why the verb form 'setting' is used. 'Setting' is a present participle, and the phrase 'setting free ..etc' is a participle phrase referring to the document (the deed of emancipation). This phrase is dependent on 'the deed of emancipation' and gives us more details about what this document was about. By the way, 'filing' makes more sense than 'filling'.

The best way I can answer this is by saying, It's just the way the verb is. you can say "setting free the slaves" or "freeing the slaves" however it is more comfortable to say "setting free" meaning he let them go to no longer work for him. I understand why it can be confusing because in a different sentence "setting" can mean something totally different "setting the table." I think it's just something you get used to hearing over time

When you put a verb ending in certain consonants into the gerund, the final consonant of the verb is often doubled. Beg -> Begging, Spar -> sparring, etc. This is to preserve the pronunciation.