Your stable will start out with 50 stalls. You need to have enough stalls to house all of your horses, so eventually you will have to expand your stable. You can do so by going to the Stable tab and clicking Expand Stable in the left side under the 'Manage' header. It's a good idea to figure out where this is! It's unlikely you will need to expand your stable until much later as those first 50 stalls will last you quite a while.
You can also see how many stalls you have on your Stable page. It will look something like "Stalls: 8/50 filled".
If you are within 5 horses of your stall limit, there will be a warning icon next to your stall count. This is just something to keep in mind as you buy more HT horses and start purchasing from the Sales Arena or Auctions as well.