First one is "to attract". "Затягивать" is "to pull smth in (some location)". So, it is also synonim for "to attract". So "life without responsibility attracts young man and he 'sinks into it and doesn't want to get out'". Imagine pit of antlion for this process.

Second one is about "to delay". "Тянуть" here is for "to pull (for example string from spool of thread)". Imagine piece of bubble gum and need to bisect it. Instead of cutting it to parts by knife you begin to pull it in two different directions (trying to divide), but it becomes string and process takes looooong time. This is why "тянуть" can be synonim for "to delay some work/process, making completition time too long". Note that "затягивать" can be used in this meaning too.