Learn Russian: Вчера я наконец смог выспаться, и теперь чувствую облегчение. - Yesterday I finally managed to get enough sleep, and now I feel relief.

The combination of the prefix вы- and the postfix -ся indicates that an action was performed until you are completely satisfied or saturated. While спать simply means to sleep, выспаться means to get enough sleep, or to sleep until you are completely rested.

Why is the perfective past tense смог used here instead of the imperfective мог?

The perfective смог specifically means that you managed to do something. You made an attempt and succeeded on this specific occasion. The imperfective мог would only mean you had the ability or opportunity to sleep, but it would not confirm whether you actually achieved the result.

Why does выспаться have to be perfective here?

After verbs denoting successful completion of a specific, single action like смог, the following infinitive needs to be perfective to focus on the achieved result. You successfully reached the state of being fully rested, so the perfective выспаться is the only correct choice.

I know the word легко (easy or light). Is it related to облегчение (relief)?