The custom sky is a feature that is not in the default Minecraft game. Optifine is an optimization mod offering many configuration options and allowing this feature to exist. You will therefore need to install Optifine in addition to the resource pack in order to run the custom sky.
Launch Minecraft in the version you want to install Optifine to get the Minecraft version on your computer. Close Minecraft.
Download the Optifine version you want. Agree to save the file despite your browser's warning; it does not represent any danger.
Go to your downloads folder, right click on the downloaded Optifine version, then run it with Java. A pop-up opens, click on Install.
Restart Minecraft with the version of Optifine you have installed. Now you have Optifine and can play with it!
Download the resource pack in the version you want and make sure it's a .zip file with a data folder, a pack.png and a pack.mcmeta file in it. If not, unzip it.
Launch Minecraft with the Optifine version you want.
Click Options, Resource Packs and Open resource pack folder to open your Minecraft resource pack folder. Drag and drop the previously downloaded resource pack in this folder and close it. Click Done and Resource Packs another time. The resource pack is now available to be selected. Select it and click Done. The textures of your game will reload and here it is: the resource pack is selected!