Silver, Gold and Platinum are amongst the best conductors of electricity, but also the amongst the most unreactive. Since electrical conductivity depends on the number of delocalized electrons (along with resistivity), if an element gives up its electrons readily (making it highly reactive), then shouldn't that mean it will be a great electrical conductor? Why isn't Cesium the best conductor?
One of the reasons why non reactive metals are good conductors is that they are good at staying as metals. Most metals react with the atmosphere to form oxides. And the majority of oxides are insulators or semiconductors. Of course there are few exceptions to this rule.
Also, just a note: calcium and iron have better conductivities than platinum.
First, why are they "non-reactive"? They're really not "non-reactive", gold and silver certainly appear in a variety of complexes, but they are highly resistant to corrosion and oxidation, which is what makes them far less reactive with typical reagents than most metals. They have this property because they have a high electronegativity (EN), so its harder for oxygen or acids/bases to remove their electrons and force them to react.
The requirements for good conductor is partially filled valence bands (alot of electrons) and strong delocalization (electrons can move freely between metal atoms). Ag, Au, Cu have alot of electrons per atom, and due to the distance from the nucleus and shielding, valence electrons aren't so tightly bound to an individual nucleaus that they can't be shared equally.
The equal sharing is key and makes this bonding is highly energetically favorable because it fills each atoms valence, but leave no Au, Ag, or Cu atom deficient of electrons. If these metals bonded covalently to a higher EN atom, they would lose some of their electron density, which is unfavorable. You'll notice however, that gold dissolves perfectly into mercury (slightly lower EN, many free electrons) because of this strong metallic bonding.
The delocalization of electrons is mainly s and p electrons (probably not what you would expect), so heavier atoms that have strong relativistic effects causing the contraction of s orbitals don't act as well as conductors because the sea of s and p electrons isn't as free. Au, Ag, Cu hit the perfect spot on the periodic table where they have many electrons, don't have a full valence (so want to form covalent bonds), but also have a higher EN than most metals so its preferable for them to bond with themselves.
An atom like Cs, has relatively few electrons to move around its metallic bonding complex, so doesn't fulfill both the requirements for a conductors even though the electrons are highly delocalized.
If you're really interested you should read about the tight binding model and the quantum mechanical approach to metallic bonding.
The simplest answer I can give is that silver (e.g) is a good conductor of electricity because of its sea of delocalised electrons between the layers of metal ions. However it is not highly reactive because you would need a lot of energy to break the metallic bonds within each of those layers.
The electrical conductivity of elements does not really depend on its reactivity. Then silver would be one of the best insulators. It actualy depends on the number of free electrons in the atoms. So, diamond has no free electrons thus it is a good insulator whereas silver having a lot of free electrons is the best conductor.