The reason why carbon dioxide is a gas and silicon dioxide is a solid is because their chemical structures are different.
Carbon dioxide is a linear structure with two double bonds between carbon and oxygen. It is a small molecule and non-polar with only weak bonds between the molecules. Hence it is a gas.
Silicon dioxide is not formed of small molecules. It consists of an infinite array of silicons where each silicon is bonded to four separate oxygens (and each oxygen is shared between two silicons). This creates a strong refractory solid (glass and sand are mostly silicon dioxide aka silica). So the same apparent overall formula doesn't describe the actual structure of the compounds at all. But the structures explain the difference in behaviour.
Of course this doesn't explain why silicon prefers to bond with four oxygens when carbon prefers just two. This is not completely simple and results from the relative bond strengths of carbon-oxygen bonds, carbon-oxygen double bonds and the equivalent bonds for silicon and oxygen. The simple version is that silicon oxygen bonds are strong relative to their double-bond equivalents whereas carbon-oxygen double bonds are strong relative to their single bond equivalents. Or, more precisely, if we could make a carbon-oxygen network solid with the equivalent structure to silica, it would tend to fall apart into carbon dioxide. If we could make silicon dioxide molecules, they would react with the release of energy to give silica.
Deeper explanations would need to look at why the relative strengths of double and single bonds turn out that way, but that would get into molecular quantum mechanics and would not be much more useful as an explanation.
The simplest explanation is the fact that the structures are different.