～ていたい is the combination of ～ている and ～たい ("want to ～"). ～ている describes the continuation of an action ("is doing") and the continuation of a state ("has done").
So 見たい is "want to watch", whereas 見ていたい is "want to be watching." Typically ～ていたい is used when you have already started something and want to continue what you are doing.
I want to watch TV.
I want to keep watching TV. / I want to watch TV more.
The difference between 壊さないでいたい and 壊したくない is somewhat obscure, but the former basically says you want to keep the current "not-broken" status.
Sometimes ～ていたい can also describe the continuation of state until some time point in the future. Using a typical instant state-change verb 結婚する ("to get married"):
I want to be married within five years. (literally "I want to have gotten married in five years" i.e., I want to be in the 'married' state in five years)
I want to marry after 5 years from now. (i.e., I don't want to marry for the time being)