Musically, what are OVERTONES, successfully introduced into pop music by the Beatles (Sgt. Pepper), The Beach Boys (Pet Sounds), and which culminated in "Monday, Monday" (Mamas and Papas) and "Close to You" (The Carpenters)?

This is only a partial answer, as I know what overtones are, but am not sure how they were specifically used in any unusual way in the songs mentioned.

Every musical note has a fundamental frequency, which identifies its pitch (how high or low it is). At the same time, the source of the sound will create other sounds, whose frequency is higher than that of the fundamental and has one of several fixed frequencies relative to the fundamental. These extra sounds that are present in the sound are called overtones. Different instruments (or other sound sources) produce complex sounds which have the various harmonics (another word for overtones whose frequency is an integer multiple of the fundamental) which are present in varying intensities. This is what gives each instrument its own characteristic sound.

Perhaps the pop music use of overtones referred to in the question refers to the introduction of 'the barbershop effect'.

In barbershop music, a style of four-part singing, the word overtone is often used in a related but particular manner. It refers to a psychoacoustic effect in which a listener hears an audible pitch that is higher than, and different from, the fundamentals of the four pitches being sung by the quartet. The barbershop singer's "overtone" is created by the interactions of the upper partial tones in each singer's note (and by sum and difference frequencies created by nonlinear interactions within the ear).

https://en.wikipedia.org/wiki/Overtone