Well...your Service Area analysis did a pretty good job of identifying roads that weren't connected.
If you're using Any Vertex connectivity, you can automatically add vertices at locations where roads cross using the Integrate tool with 0 XY tolerance. Make a backup copy of your data first...
If you're using End Point connectivity, you'll have to split features at intersection points, which can be done. However, if you have fields in your data like address ranges or travel times, keep in mind that these could get messed up if you start splitting features. You'll need to make sure that your geodatabase domains are set up with the appropriate merge/split policies.