Tim Bowen puts his heart into these heart-related idioms.
A performer or sports person who plays their heart out is said to have put the maximum amount of effort into their performance. This idiom is also often used with the verb sing. Even if their performance turns out not to be as good as expected, they can take heart from (feel encouraged by) the knowledge that they did their best and they shouldn't lose heart (feel disappointed and try less hard). If they do this, then next time their performance will really show that their heart's not in it (they don't really care). However, if a performer has obviously tried hard, then the audience would have to have a heart of stone (be extremely unsympathetic) not to applaud them for their effort in the end.
When it comes to sports games, fans often set their hearts on (decide they want very much) their team winning, and if the game is exciting they may have their hearts in their mouths (feel extremely apprehensive and worried) throughout. If their team ends up losing, it can cause their heart to sink (feel very disappointed), even if in their heart of hearts (their secret feelings) they knew they were going to lose. Again, if the team has tried hard, people may say their heart bleeds for them. Taken at face value, this expression means that you feel sympathy for someone, but it is usually used sarcastically to indicate the opposite - that you do not feel any sympathy for them whatsoever.