Translation:I cannot relax!
放心 means don't worry.
I think it's easier by understanding the meaning of characters: 放 is an action, it alone does not explain much except an action. 松 means loose. 放松 means an action to loosen, hence the English translation of relax. 心 means heart. 放心 means an action to put down your heart, which we normally mean "not worried".
From my interpretation of the dictionary entries, seems that 休息 is to rest (from a state of work)/to take a break from something; 轻松 is to take it easy/to relax/the state of being relaxed; 放松 is more associated with the act of becoming relaxed/loosening (from a tense state).
For more context can look at rough translations of the individual characters. Both of 休息 separately ~ rest. 松 ~ loosen/relax. 轻 ~ light/easy/unstressed. 放 ~ release/let go. So 轻松 ~ relax and take it easy; 放松 ~ let go of stress.