When someone says that their heart soars, it simply means that they are really happy. Her heart soared (happiness and excitement) when learning of her pregnancy. His heart sank (sadness and disappointment) when he heard he didn't get the job.

soar (flying high-up)

sink (falling-down)

I don't think this is a great sentence though.  I think it would be better replacing 'with' with 'at'.

When a bird "soars," it flies higher and higher without flapping its wings. When we watch, we often experience a feeling of joy because it looks so graceful and effortless.

Figuratively, we say our emotions "soar" when we experience an effortless, spontaneous lifting of our spirits.

If I hear that a predicted snowfall has taken a different path than forecast, and will miss us, I feel "relief."

But when I see my grandchildren doing something cute, my spirits "soar."