该 is softer than 必须 (must) but stronger than 可以 (may). It implies something is expected or proper.
usage
该...了 pattern expresses 'it's time to...': 该吃饭了 = It's time to eat, 该走了 = It's time to go.
register
In formal writing and the news, 该 also points back to something just named, like 'the said' or 'that': 该公司 (the company), 该国 (that country). In speech people use 这 or 那 instead.
Left speech radical (simplified from 言) - the indexing semantic. Heads verbs of speaking, agreeing, promising, judging: 说, 话, 谁, 谢. In 该 it points to the verbal sense of stating an obligation: words about what someone ought to do.
Right 亥 gives the sound - shares the rime with 该 gāi, with the initial shifting (h→g) but the rime preserved. A typical phonetic compound. Same phonetic body produces 孩 (child) and 骸 (skeleton); the 'pig / earthly branch' meaning of 亥 doesn't carry.
