I'm going to attempt an intuitive explanation in a similar style to the linked question:

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>First of all why not simply exclude the uncertain data all together?

There might be several reasons for this. One might be interested in computing quantities that require that value be present, for example a regression or classification model. In these cases, "imputing" $z$ in this manner is more principled than picking an arbitrary value, particularly if the values of $x$ and $y$ might feasibly have something to do with why $z$ is missing.

This is (I believe) what the linked question in your linked question was discussing

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>Now in order to use this data and make GP predictions on some new location (xnew,ynew) to obtain znew we must take into account all the uncertainty related to z∗, which now has pdf due to its uncertainty, and integrate it out!

Just to clarify a potential source of confusion here: the uncertainty that we "integrate out" is the *predictive* uncertainty associated with predicting $z_{\mathrm{new}}$; that is, for new points $(x_\mathrm{new}, y_\mathrm{new})$, we predict the unobserved $z_\mathrm{new}$, which *due to this prediction*, is uncertain.

Any uncertainty related to $z$ (that is; the observations we *do* have, *not* ones we might want to predict) should be included in the GP we use to model the *unobserved* $z_\mathrm{new}$. 

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>How does integrating or summing up all the values of uncertain $z^∗$'s  help us as described in the previous post?

Essentially what's happening here is we're taking into account the fact that $z^*$ has to be *predicted* and thus is fundamentally uncertain. This integration is an attempt at accounting for this prediction uncertainty in whatever it is that's being predicted.