The following is a good example of Indo-Europeanists gone mad, methinks. Whenever it comes to European vocabulary, specialists of Indo-European languages are in there like a dirty shirt trying to etymologize it automatically through some concocted Indo-European root. Some roots are valid naturally, but others are suspicious and crave our skepticism.
Specialization can sometimes lead to shortsightedness. We can see why not all words we find in European languages are necessarily inherited from a language spoken 6000 years ago. There are many sources of vocabulary in any language and to assume a priori that words are automatically inherited is the same fallacious tactic used by zealous Nostraticists to claim an overabundant number of terms have some great heritage that never existed. It's much more likely that any given word, when going back six millenia, is a borrowing rather than an untouched, ancient word. Yet IEists, because of their passionate obsession, risk being ignorant of the contribution that must have also been made by Semitic, Egyptian, Basque, Aegean and other language groups. They may often stick to what they know, namely Proto-Indo-European, and attribute everything to it alone.
Enter the etymology of Capua, a city in Italy which was originally controlled by Etruscans. The Latin name was identical and called in Greek Καπύη (Kapuē). Charlton Lewis, to which this reference in Perseus owes origin, went straight to the punch and pointed the reader to Latin campus meaning 'field'. When we next look up campus, we're led to a further connection with Greek κῆπος (Doric κᾶπος).
This is interesting because further investigation reveals contradictions about the etymology of these words among IEists. Mallory and Adams have attributed κῆπος to an Indo-European root *ḱāpos 'garden' while Pokorny had claimed a different root *kam-p- 'to bend' for the Latin reflex while dragging in another Greek word καμπή 'a winding (of a river)' as alleged cognate. I believe both reconstructions are likely incorrect. Pokorny's is easiest to dismiss since the relationship between campus and κῆπος is direct, phonetically and semantically, yet his etymology would absurdly contradict this in favour of 'bending' the meaning of these words to suit a nuance that isn't attested. M&A's reconstruction at least acknowledges the core meaning in these words, 'cultivated land', but then, despite this, the connection with Capua and Campania put forth in Lewis' work is left unaddressed but worthy of a solution. We're also offered a highly unlikely PIE root with a long low vowel *ā which is odd considering that even short *a is rarely attested in PIE. Should we additionally assume a laryngeal before *p? Or perhaps should we accept the problems of these attempts to source these words to PIE and try a fresh approach?
Let's look outside the PIE bubble for argument's sake and try this from an Aegean perspective. I want to suggest an Etruscan word *capa and assign it the meaning of 'field'. This word then can serve as a direct source for Latin campus with homorganic nasal resonant -m- added. As an inanimate noun, the Etruscan plural is predictably *capava 'fields'. Coincidently, this is precisely the name of Capua in Etruscan, attested in the locative case as Capue (TLE 890) (from Old Etruscan *Capava-i). The citation form, *Capava, would have effectively meant 'The Fields', referring to the scenic topology of the area. After solving the meaning and source of Capua, we can push on to reconstruct the Etruscan name for Capua's surrounding region, *Capavana 'Campania', economically solving for the source of the Latin name too. Finally, this opens doors in regards to the exact source of the Greek words for 'garden', ie. 'Pelasgian' substrate, an idea already suspected by Robert Beekes.
 Mallory/Adams, The Oxford introduction to Proto-Indo-European and the Proto-Indo-European world (2006), p.163 (see link).