I started a stable and have some space for new horses, but I am puzzled on how to attract new owners to place horses to my stable? I have a middle rated performance and held several open days.

Other owners will not put horses into your stable. You have to buy horses in auction or breed them yourself.

Does it apply to trainer only mode? I am not saying both trainer and owner mode. I did breed horses but I don't know how to buy horses.

As you click skip through the days every few you should see a coloured circle either red green or yellow. Click on it and it will show you all horses up for auction. Bottom right of screen is a button to start the auction. You then have to bid to compete with other trainers or that horse.

You cant buy horses in trainer only mode, At the start of every season you should get some owners to move there horses in with you.

make sure you are impressing the owners, winning races etc.

In the stable Upgrades page is an "Open House". Helps to do that every so often.

Yes I held open days, but not all time have a return. But just got a big name with generous support.

Get Elton John singing at the open day that will draw the punters in

What you need to do in trainer mode is get your stable facilities as good as possible. Difficult as you only start with 100k and generally poor horses, so you have to gamble your way to new facilities. Once you have a really good stable you will get offers from owners sometimes moving their entire string to you 80/90 horses! you will get a lot of good horses when you get the big owners but wow it takes some managing!! once you get to this stage you will be wiping the small owners off the bottom of your shoe. Just remember it is all about the facilities

Best wishes
Martin