How does a reduction in consumer spending in favour of consumer saving, affect economic activity? Let's say our government runs an advertising campaign, encouraging its citizens to reduce spending money on consumer items, and instead save more money. 

Let's say the campaign is successful, and that happens.

How does this affect economic activity? 

On one hand the factory and shop owners appear to produce less, and thus hire less people, but on the otherhand, there is more money available to build new factories etc.
Such a "planned" and sought-after re-allocation of given income from consumption to saving, is justified only if the savings in an economy are sub-optimal (or we think so), in the sense of hurting the investment rate, which in turns hurts the (human and physical) capital infrastructure.  

Think about the extremes: consume all that you produce, save nothing (as a society). Your capital base will erode, leaving in the end only the other factor of production, labor, an orphan, without the necessary capital to produce, leaving perhaps only land for the society to return to a non-industrial agricultural state.  

So it is a matter of intertemporal (re)allocation of resources in order to guarantee that the economy survives, and then that it does a bit better than just surviving.

The conundrum: if businesses see consumption declining, why would they rush to make new investments even though now the "price" of these investments is lower, due to increased savings and so to increased availability of funds?  

I would invoke here "Say's Law": Supply finds its own demand. (see the controversies over it).
For me, this law is better interpreted in the context of heterogeneity: no matter what overall consumer spending does, there is always some fields where consumer desire goes unfulfilled. Entrepreneurs try to identify these fields, and invest there (this has a Schumpeterian flavor).

The above discussion does not mean that we will not observe phenomena like increased unemployment. After all, investing in new fields requires re-allocation of productive resources, and this is not easily nor quickly done. And there is always the case that the "savings drive" may overshoot, and the economy will be led to a lengthy recession, Japan some 20 years ago being the classic modern example (in fact in Japan, the government tried actively to persuade the citizens to raise their consumption, but it failed).