Learn English – What does “Thundering typhoons” mean

It usually implies that something is little in one way but big in another; for example, a game may want to imply that it is "little" in being approachable and friendly, but "big" in having lots of things to do.

It means it's very easy. Japanesey, I would guess, for no better reason than that, like peasy, it rhymes with easy.

additional info from Magy:

James Whitmore, who played Brooks Hatlen character (and who said "easy peasy, japanesey"), was very well known actor. Digging a bit deeper in history of this actor revealed his serving with the Marines in World War II and such experience he employed in numerous movies from that period and storylines: as Vice Adm. William F. 'Bull' Halsey Jr. in famous Tora! Tora! Tora!, then in Nobody's Perfect, The Deep Six(1958), Above and Beyond(1952) and others. It's highly likely, this speech was Whitmore own contribution into movie, to give more direct and realistic feel of his character. There is also possible, that's phrase/slang was used amongs Navy crews during WWII.