Learn French: Je sais qu'il va démissionner. - I know that he is going to resign.

In French, the conjunction que must drop its e and add an apostrophe when the following word starts with a vowel or a silent h. Since il starts with an i, que il must become qu'il.

Can I leave out qu' like we do in English and just say "Je sais il va démissionner"?

No, you can never drop it in French! In English, it's very common to leave out the word "that" and say "I know he is going to resign." In French, you must always include que (or qu') to connect the main clause (Je sais) to the subordinate clause (il va démissionner).

Why do we use savoir (Je sais) here instead of connaître? Don't they both mean "to know"?

They do, but they are used in different situations. You use savoir for knowing facts, knowing how to do something, or whenever a phrase is introduced by que (that). You use connaître for being familiar with a person, place, or thing (like "I know Paris"). Because we are stating a fact connected by que, savoir is the only correct choice here.

What tense is va démissionner?