What is the meaning of "Many a mickle makes a muckle"?

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’.