This quotation comes from Jaques's famous speech, beginning "all the world's a stage." The speech describes the so-called "seven ages of man," moving from infancy through youth and adulthood to the final stage of existence, which is described as a "second childishness."

Of course, this is not literally a second childhood but rather extreme old age. Jaques is referring to the fact that many people in extreme old age shrink to a smaller size, and their voices become "treble" again, losing their strength and becoming quavery and more high-pitched. The reference to a second childhood also seems to allude to a slide into dementia and the inability to care for oneself, particularly when afflicted with senility. In the final line of his speech, Jaques explains some of the other elements which connect the very old person to the child: like an infant, very old people may find themselves unable to see, without teeth, and unable to taste anything. They enter into a sort of "oblivion" as they cease being able to properly understand the world and become dependent on those around them for survival, as a very small baby might be.

Obviously, Jaques is here painting an extreme example of the very elderly. It is not always the case that the very old become senile, lose their minds or senses, and become completely dependent on others. But in the most extreme form, old age can indeed mimic childhood in many ways.

JAQUES. Last scene of all,
That ends this strange eventful history,
Is second childishness and mere oblivion;
Sans teeth, sans eyes, sans taste, sans everything (2.7.170–173).

In what is essentially a "filler" speech in act 2, scene 7 of William Shakespeare's As You Like It—used to pass the time between Orlando going offstage and returning with Adam, his aging servant—a cynical, melancholy courtier named Jaques gives a two-minute speech which is famously known as "the seven ages of man."

The speech begins with " All the world's a stage, / And all the men and women merely players" (2.7.146–147) and ends with the lines cited above.

"Second childishness," or extreme old age, is often analyzed and interpreted negatively, most often in terms of the words that follow—"mere oblivion" and "Sans teeth, sans eyes, sans taste, sans everything." Nevertheless, "childishness" can also be interpreted in terms of its alternative, positive definition of being "childlike," and as distinct from the utter helplessness and dependence on others that Jaques implies is the last stage of life.

While it's true that the youngest of children have no teeth, the same cannot be said of a child's sight, taste, or any other of its senses, through which the child discovers and experiences itself and the world around it with excitement and wide-eyed wonder.

The youngest of children also require constant care, as do some of the oldest among us, but it's not long after a child learns to speak that it demands to "do it myself." The same is true of some of the oldest among us who resent having things done for them, particularly if they believe that they're capable of taking care of themselves, even if in a limited capacity.

In his cynicism, Jaques sees these as negative qualities, in much the same way that he focuses on the negative aspects of each of the "seven ages"—the "mewling and puking" of the infant, "the whining school-boy," the "woeful ballad" of the lover, the soldier who is "quick in quarrel," seeking only the "bubble reputation," the justice who spouts profound nonsense, and the "pantaloon," the foolish old man with a "childish" voice whose tights are too big for his skinny legs.

Jacques's use of "childish treble" in describing an old person's voice is a clue to the meaning of "second childishness," a phrase Jaques uses just three lines later in the speech. This same "childishness" is exemplified by Jacques himself in the childlike excitement of discovery he feels in his "A fool, a fool! I met a fool i' th' forest" speech earlier in the same scene (2.7.12–35, 37–44).

In the monologue of Jaques from Shakespeare's "As You Like It," the borther to Orlando capitalizes on the words of Duke Senior who states that "we are not all alone unhappy...in this wide and universal theatre."  Jaques calls the world a stage upon which men and women are players, making their entrances and exits.  In the first stage, the infant is "mewling and puking in his nurse's arms."  Finally, in the seventh stage, man has returned to the infantile stage as he exists in "mere oblivion" and is "sans"/without teeth, without sight, without taste, without everything.  Like babies who have yet to reach the development of their senses, the aged have their senses mitigated to a similar point, but it is the loss rather than the nascence as in infants.

In this poem, Shakespeare uses the phrase "second childishness" to refer to the way people are when they get old.  This used to be a very common euphemism for senility -- for the way that some people end up when they become old.

Some old people end up more or less like children.  They have less control over their emotions and are less inhibited.  This makes them act like children.  In extreme cases, they lose control of their bodies and need people to help them get dressed, wash themselves, etc.

Because some old people end up this way, the term is often used (not as much today) as a way of referring to old age.