What is a better way of saying this sentence?
The banker explained the working of bank in an easy way for a layman to understand without any core banking words.
The emphasis is on the highlighted part.
..without any banking industry jargon
Jargon's definition from dictionary.reference.com/browse/jargon
"The language, especially the vocabulary, peculiar to a particular trade, profession, or group"
If the emphasis is meant to be on the bolded words (that is, the avoidance of jargon is the key message to be conveyed) then you could move it to a more prominent position.
"The banker explained how a bank works in layman's terms"
"The banker explained in layman's terms how a bank works"
are shorter and incorporate the highlighted portion of your original sentence in the phrase in layman's terms.