There are a few ways to spend your Apex Coins: on new Legends, Legend Skins, Apex Packs, and the Battle Pass.
If you want to use your Apex Coins to purchase new Legends or Legend Skins, head over to the Store tab in the lobby. Here, you'll see all of the different Legends that are available for purchase, as well as any Legend Skins that are currently for sale. If you see something you like, simply click on it and select Purchase with Apex Coins.
Apex Packs are loot boxes that can be bought with either Credits or Apex Coins. These packs guarantee at least one Legendary item per pack, and they can contain a variety of other items such as skins, voice lines, banner poses, and crafting materials. You can find the option to buy an Apex Pack under the Store tab in the lobby; just look for the Apex Packs section.
Finally, if you want to use your Apex Coins to purchase the Battle Pass (which gives you access to exclusive cosmetic items), go to the Battle Pass tab in the lobby. Here, you'll see how many Apex Coins is required to buy the pass; once you have enough coins, simply click on Purchase and follow the prompts.