TRY BELOW SOLUTIONS.
SOLUTION 1:
-> Run Android Studio as Administrator (this is only required for first run or when any issues arise due to automatic updates on any run in future).
Though it shows API 23 is or any other updates are not installed (and wouldn't show any 'Retry' button), BUT still the download and installation of rest of the updates would proceed (you can see the download progress incrementing even after seeing the above errors). So, DO NOT ABORT the operation.
-> At the end of installation (after rest of successful updates are installed) it would this time display the "Retry" button to retry installation for API 23 or any other versions which had failed earlier. Then click the "Retry" button, it would work this time.
This would resolve your issue with successful installation.
-> Next time onward no need to run Android studio as Administrator, unless it does any automatic updates and shows similar issues.
SOLUTION 2:
-> Alternative approach is to install any updates (which ever failed in earlier attempt) from Android SDK Manager first and then later launch Android Studio (which would not need any check for updates and any additional installations).
To install any features or updates, run Android SDK manager as administrator (run as administrator is not mandatory for this but preferred to avoid any permissions related issues) and check the required options and proceed with the installation without any issues.
SOLUTION 3:
-> If still your issue is not resolved, then try the proxy solution as suggested by others or check your internet connectivity if it's working properly or not.
*Any of the above solutions should resolve your issues.