I've heard this phrase, and don't know what a "mickle" or a "muckle" is. Hence I have no idea at all what the phrase itself is supposed to mean.
In this phrase, a mickle is a small amount of something (the Scots usage is intended in this proverb) and a muckle is a large amount, so the saying means that you can accumulate a great deal by many small savings.
Some confusion may be caused by the fact that a mickle can also mean a large amount (isn't there a question about words than mean the opposite of themselves somewhere?).
"Mickle" is a (now obsolete except in dialect) word meaning "great", and is cognate with "much". "Muckle" is a variant, particularly used in Scotland.
The OED says of the phrase you are asking about:
[mickle, n.:] A large sum or amount. Chiefly in proverb: many a little (also pickle) makes a mickle (now freq. in the garbled form many a mickle makes a muckle).
The form many a mickle makes a muckle (earliest recorded in quot. 1793) arises from a misapprehension that, rather than being variants of the same word, mickle and muckle have opposite meanings, the former representing ‘a small amount’ and the latter ‘a large amount’.
A Mickle is an Irish word for 'coin' and a Muckle is an old Cockney term (derived from old Yiddish slang) meaning a 'bundle'. The phrase means to save each coin and create a bundle, i.e. a bundle of coins. In more modetn venacular it translates to 'save a penny, save a pound'.
Keith Waterhouse did not invent the term but attempted to relate it to a wider audience - being the literate classes as the folk who actually used the term were illiterate working poor of London. He was using it to relay amusement at the kind of expressions used by the uneducated poor. In addition, though the phrase was used up North in Yorkshire and Scotland it originated in the East End of London.
Interestingly enough 'to take the mick' is from the same Irish word 'mickle' - to take the 'mickle'/ to take the 'penny' (i.e. to fool or deceive someone of their money). Also you will still sometimes hear a London Black Cab driver use the word 'Muckle' to describe a large amount, for instance 'theres a Muckle to be made tonight' or 'theres a Muckle to be made if you do the hours', however disclosure of good earnings is usually only discussed in the confidence of another Cabbie.
Stand on me, I'm a 3rd generation Cabbie.
"There's many a mickle makes a muckle" is a spoof - it has no meaning. It is a saying invented by two characters in the Keith Waterhouse book Billy Liar to parody Councillor Duxbury, the co-founder and boss of Chadrack and Duxbury funeral service. Duxbury has a range of obscure Yorkshire sayings that are a source of amusement to his younger, less reverential, employees. They go on to invent some sayings which sound plausible but are meaningless - "I'm about thraiped wi' thee" or "There's many a mickle makes a muckle".