Come gather round, a story I have to tell.
It should be familiar, you ought know well
History - which has (i.e. contains) the word 'story' within it - is something you study in school, hence it is familiar and something everybody who has ever been to school should be aware of.
Of things past, I give life anew
History concerns things that have already happened - it is the study of the past. Talking of history can metaphorically make you feel like you are bringing events from long ago back to life.
Of things to come, I know not false from true
It is not - and cannot be - the study of the future; it knows nothing about things that have not happned yet...
Who am I you say, I ask you the same
If you want to know who someone from the past was, you can generally find out as the things they did, where they lived, etc. are all preserved by history - books, encyclopedias, census records, birth/death/marriage registers, the Internet... History asks 'who was this person?' and also knows the answer!
Speak my name right, the answer you shall claim
The word 'history' sounds like 'his story', which is precisely what the narrator is telling us!
As for the title:
'The spectre of history' is a common phrase, often used with 'looms large' when somebody has a lot to live up to because of their predecessors' past achievements or when things that happened in the past continue to haunt them and appear to prevent them achieving something... (Think every Manchester United manager after Alex Ferguson!)