If carbon–carbon bonds are reasonably strong and silicon and carbon are both in the same column of the periodic table meaning they have the same amount of valence electrons, also seeing as its bonding with itself electronegativity isn't an issue, why are silicon-silicon bonds weak?
At the risk of oversimplifying: bonds form because the bonding electrons can lower their energy by being attracted to both nuclei. Take the simplest possible example of H$_2$. The electron density looks like:
The formation of the H-H bond increases the electron density between the two protons, and the electrons in this region have a lower energy because they feel the attractive force of both protons.
If you now consider a bigger atom, then the inner electron shells that don't participate in the bond will increase the size of the atom and therefore increase the distance between the nuclei. Because the electrons in the bonding region are farther from the two nuclei the energy gain they obtain from being in the bond is less, so the bond is weaker.
There aren't any higher atomic weight analogues of Hydrogen, but consider the halogens instead as they form dimeric molecules. If you look at the formation enthalpy of the X-X molecule it varies down the group, i.e. with increasing atomic size, as:
Apart from Fluorine (see below) you see the bond enthalpy decreasing with increasing atomic size.
The reason Fluorine is anomalous is because the F atom is so small that the lone pair electrons get crowded in F$_2$ and decrease the bond strength. If you look at the HX molecule instead you don't get this effect (because H doesn't have any lone pairs) and you see the trend you expect:
And this applies to carbon and silicon as well. The reason the Si-Si bond is weaker than C-C is simply that because the Si atom is bigger than the C atom the electrons forming the bond are farther from the nuclei and don't lower their energy as much by forming the bond.